UNIVERSITY OF CINCINNATI

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1 UNIVERSITY OF CINCINNATI DATE: August 14, 2002 I, Manuel Valera, hereby submit this as part of the requirements for the degree of: DOCTORATE OF PHILOSOPHY (Ph.D.) in: Physics It is entitled: Density Functional Study of Classical Liquids Approved by: Dr. Frank Pinski Dr. Rohana Wijewardhana Dr. Howard Jackson Dr. Fu-Chun Zhang

2 DENSITY FUNCTIONAL STUDY OF CLASSICAL LIQUIDS A dissertation submitted to the Division of Research and Advanced Studies of the University of Cincinnati In partial fulfillment of the requirements for the degree of DOCTORATE OF PHILOSOPHY (Ph.D.) In the Department of Physics of the College of Arts and Sciences 2002 by Manuel Valera B.S., Simon Bolivar University, 1992 M.S., University of Cincinnati, 1996 Committee Chair: Frank J.Pinski

3 Abstract A study of the freezing properties of simple liquids composed of hard and soft spheres is performed using classical density functional theory (DFT). The systems studied are composed of single component and binary mixtures. The behavior of binary mixtures as a function of the size ratio is analyzed. DFT equations are solved using an enhanced numerical method, which is based on a real-space mesh that does not constrain the shape of the density. Fast Fourier transforms are used to produce solutions efficiently. The reliability of the method is studied as a function of the mesh size in order to find an optimal implementation. A comparison with the exact known molecular dynamics result for single component systems shows excellent agreement. The solutions for binary mixtures of hard spheres reveal a qualitatively different behavior from previous results obtained using plane waves and gaussian parametrization. An fcc structure is found for almost identical particles, size ratio ~ 1:1. As the size ratio is reduced, a sublattice-melt phase is found with the small particles having a nonlocalized density and a multiple peak structure. Finally, the system reaches a NaCl structure at a size ratio of (0.45:1), with well-localized peaks in the density for small and large particles. For the same range of size ratios, the study of soft spheres mixtures reveals a similar behavior with the existence of non-stable solutions for size ratios of (0.80:1). Calculations on ternary systems were performed and the existence of the critical polydispersivity was found, which suggest similarities to real polydisperse systems.

4 Acknowledgments I would like to express my appreciation to my dissertation advisor, Dr. Frank Pinski, for all his guidance, support and patience during the course of my studies, which enabled me to complete my research work. I would like to thank Professor Duane Johnson for his support during my research period at Sandia National Labs, and his continuing advice after my return to Cincinnati. I would also like to thank Dr. Bulbul Chakraborty for many helpful discussions. Thanks to my mother, my father and my family for having faith in me. Finally, I would like to thank my wife, Ingrid, for her constant help and patience during the final years of my thesis work.

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6 Table of Contents LIST OF FIGURES... 3 LIST OF TABLES... 5 CHAPTER I INTRODUCTION... 6 CHAPTER II THEORETICAL BACKGROUND II.1 Density Functional Theory II.1.1 General Theory II.1.2 SecondOrder Theory of Freezing II.1.3 Weighted Density Functionals II.2 Integral Equation Theory of Liquids II.2.1 Pair Correlation Function II.2.2 Structural and Thermodynamics Properties II.2.3 Integral Equation Approach II Hypernetted Chain Equation II Percus-Yevick Equation II Rogers and Young Equation II.2.4 Generalization to Mixtures II.3 Hard-Sphere and Soft-Sphere Potentials II.3.1 Hard Spheres II.3.2 Soft Spheres CHAPTER III METHOD OF SOLUTIONS III.1 Method of Solution of Density Functional III.1.1 Review of Popular Methods III.1.2 Discrete Method III.2 Method of Solution of Integral Equations CHAPTER IV RESULTS AND DISCUSSION IV.1 Density Functionals

7 IV.1.1 Test of the Discrete Method IV Comparison for Single Component Systems IV Study of the Grand Potential as Function of the Grid Size IV.1.2 Freezing of Binary Hard Spheres IV Calculations Performed IV Results for Binary Systems of Hard Spheres IV.1.3 Freezing of Binary Soft Spheres IV.1.4 Comparison of Results of Soft and Hard Spheres IV.1.5 Polydisperse Systems and Ternaries IV.1.6 Glassy State IV.2 Calculation of the Diffusion Coefficient Using Integral Equations CHAPTER V CONCLUSION BIBLIOGRAPHY

8 List of Figures Figure 1. Graphical representation of the partitioning of the unit cell used in the discrete method Figure 2. Grand Potential Difference for a single component hard-sphere system obtained using the discrete method is plotted. The different curves represent results for different grid sizes Figure 3. Liquid and Solid density for hard-sphere mixtures are plotted as function of the size ratio. The solid coexists with the equimolar liquid Figure 4. Contour plot of the density of large spheres is displayed for a binary mixture of hard spheres for size ratio of The area shown is the face of a unit cell. The fcc structure is evident from the positions of the peaks in the density Figure 5. Contour plots for hard sphere binary mixtures are displayed for various size ratios. The area shown is the face of a unit cell. The fcc structure is evident from the positions of the peaks in the density of the large particles Figure 6. Density profiles of the small particles are displayed for binary hard spheres at freezing. The solid is coexisting with the equimolar liquid Figure 7. A plot of the fluid and solid coexistence densities of large spheres as 3

9 a function of the size ratio for different crystal structures in coexistence with a fluid concentration of Figure 8. The Grand Potential Difference plotted as function of the lattice constant for binary soft sphere systems. Plots for σ 2 / σ 1 =0.92, 0.90, 0.89, 0.88 are shown. For each size ratio, curves for densities close to the freezing point are shown Figure 9. Contour plots are shown for the inhomogeneous phase for soft-sphere binary mixtures with varying size ratios Figure 10. Density profiles are displayed for hard-sphere and soft-sphere systems. These are the plots for the density of the larger particles Figure 11. Comparison of density profiles for hard-sphere and soft-sphere systems. These are the plots for the density of the smaller particles Figure 12. Direct correlation function is plotted as a function of the distance for various values of n, the softness of the potential Figure 13. Density profiles are displayed for ternary systems of hard spheres Figure 14. The diffusion coefficient is plotted as a function of the reduced parameter for binary systems of soft sphere calculations. Results from MD and integral equations are shown

10 List of Tables Table 1. Solid and liquid densities ( ρ and ρ respectively) at coexistence are s shown for single component hard-sphere systems (in reduced units 3 ρσ ) l Table 2. A listing of certain thermodynamic values for hard spheres mixtures. The solid is in coexistence with an equimolar liquid Table 3. Comparison of results obtained using the discrete method with the gaussian approximation. The solid is in equilibrium with the liquid composed of equal amounts of both types of hard spheres Table 4. A listing of thermodynamic parameters for binary soft sphere systems. The solid coexists with the equimolar liquid Table 5. A list of various properties for hard-sphere and soft-sphere systems

11 Chapter I Introduction Complex systems such as gases, liquids and solids and how they transform into one another, play a significant role in the events of everyday life. We are exposed to transformations such as solidification (freezing) in many different ways. Solids, either amorphous or crystalline, find many applications in the industry, from window glasses, optical fibers and photovoltaic cells, to almost perfect crystals with wonderful electric and magnetic properties 1. There is no argument over how important such changes of phases are; yet there is little understanding of the processes and conditions that lead to these transformations or phase transitions. All substances when placed under proper conditions freeze into crystal, quasicystal or amorphous states. The physics behind this phenomenon remains somewhat mysterious. Even helium becomes solid under pressure. One such phenomenon, where we lack satisfactory understanding, is the solidification of systems into disordered states such as glassy systems. No single theory can explain this phenomenon or predict behavior based on simple assumptions 1. Different approaches have been used to study liquids, solids, glasses and their freezing properties from a classical point of view. Various methods have been applied to a wide range of systems with various degrees of success. Among the most popular theories, we should mention Molecular Dynamics 2, Mode Coupling Theory 3, Integral Equations of Liquids 4, and Classical Density Functional Theory 5. These theories have been used to understand simple systems such as hard spheres as well as more complex ones such as 6

12 polymers. Molecular Dynamics has been very successful in explaining simple systems and, in some cases, is regarded as the most accurate solution, being used as a reference system to test mean field approaches. Its implementation becomes difficult as the complexity of the system increases as well as being computationally expensive. A mean field approach, based on the assumption that short-range fluctuations have little effect on the thermodynamics state of the system, produces a satisfactory explanation with qualitative, and sometimes quantitative accuracy, while requiring less computational resources. Among the mean field class of theories, the Integral Equations approach must be considered as one of the key players 4. It has been used extensively for different kinds of systems including single component and mixtures of hard spheres, soft spheres and liquid metals. In general, such Integral Equations are relatively easy to implement and the resulting description is accurate for the gaseous and liquid states, and in some cases for the supercooled state leading to a possible description of the freezing transition 4. I will illustrate later that it provides an inaccurate description of the supercooled state for binary mixtures of soft spheres, which suggests that it may not be a suitable choice for studying systems in such regimes. A second class of theories in the list is based on Classical Density Functional Theory 5. These theories have been used successfully to explain crystallization of simple systems such as hard spheres for both single component systems and mixtures. Classical Density Functional Theory incorporates the mean field idea that the properties of the solid can be obtained by extrapolating the liquid properties into the solid regime. The pioneering work of Ramakrishnan and Yusouff 6 describes the freezing of hard spheres with great accuracy by using only a second order approximation. This work led to a new set of more 7

13 sophisticated theories that have been used to study more complex systems ranging from binary mixtures to polymers 5. The representation or parametrization of the density is a key to obtaining a good description of the system using Classical Density Functional Theory (DFT). Most implementations of Density Functional Theories rely on a given parametrization of the density of the particular system, which impose restrictions on the space of solutions that we can explore. For instance, a parametrization based on gaussian functions would only produce a density that would necessarily possess classic symmetry, i.e. spherically distributed about the lattice site. These solutions are an adequate description for single component crystals but fail to reproduce more detailed features introduced by increased complexity, as in the case of binary mixtures, which will be explained later in this thesis. In this work, I present solutions of Density Functional Theories for binary mixtures of hard and soft spheres using a density parametrization that is more accurate than the previously used gaussian methods. I demonstrate that the solutions are very different for mixtures leading to features in the density that have not been seen previously. The method used is similar to the one proposed by Dasgupta 7 and used extensively to study dynamics properties of the system. I have enhanced the method by increasing the degrees of freedom through the use of very fine grids and allowing the volume of the super cell to change. In previous work, the lack of this flexibility imposed a harsh restriction on the phase space leading to false solutions 7. The resulting coarseness of the grid leads to numerically-induced, unphysical solutions of the DFT equations. Here I must state a few sentences about hard-sphere and soft-sphere systems. The complexity of some real systems such as water or silica requires that we build simplified 8

14 models retaining the fundamental physics that can be studied and treated with the current theories. These simple models in many cases provide very good descriptions of real liquids and solids 2. They are in general isotropic liquids composed of spherical particles and subject to symmetric interactions. Hard spheres are such an example. Studying such simple systems is instructive. First, they provide a better understanding of real systems, giving in some cases a good qualitative description 4. Second, even though they are simplified models, they possess many features of real systems and undergo physical processes such as phase transitions. Thus they are an ideal tool for understanding these phenomena which are universal to all such systems. In addition, the study of hard sphere systems will allow one to isolate entropic effects from those originating through particle interactions. I will demonstrate that the refinement of the grid method that I proposed, while reproducing previous results for single component systems of hard and soft spheres, is able to obtain solutions with lower energies and a richer set of features. For binary mixtures, I will show that the structure of the stable inhomogeneous solutions differs from those previously found 5. This manuscript is structured as follows: The first chapter presents the theoretical background. In this chapter, I present a review of the fundamentals of DFT, Integral Equation of Liquids, and an introduction to both hard-sphere and soft-sphere systems. In the second chapter, I present the method of solutions for both systems, describing the grid method I used to obtain the solution of DFT equations and the iterative method used to solve Integral Equations. In the third chapter, I present the results for binary mixtures of hard and soft spheres as well as some applications of Integral Equation to glasses and DFT results for various system including ternaries of hard spheres. Some suggestions for future work are discussed in this section. Finally I present some conclusions. 9

15 Chapter II Theoretical Background Classical Density Functional Theory and the theory of Integral Equations of Liquids are two interrelated topics which, when used effectively, produce a large amount of information about the physics of many systems. In this chapter, I review the theoretical basis of these two central topics of this thesis, as well as define and derive the physical quantities used. Both theories have been used extensively to study fluids, solids and phenomena such as solidification and crystal formation. It is the purpose of this chapter to expose the relevant points and how these two topics are connected, in particular through the direct correlation function. It is not my intent to give an extensive exposition since there is a wide range of excellent reviews on this matter 4,5. Finally, I will define the potentials used in the calculations presented in this thesis, namely, hard spheres and soft spheres. II.1 Density Functional Theory In this section, I present a summary of the key points of Density Functional Theory. The second order truncation in the density is explored. The extension to mixtures is explained, followed by a brief discussion of more sophisticated approximations based on the concept of effective densities. 10

16 II.1.1 General Theory Over the past years, Density Functional Theory (DFT) has become a popular tool in the study of liquid to solid phase transitions and other phenomena involving non-uniform classical liquids 5. This is due, in part, to the success obtained in the description of the fluid-to-solid transition of hard-sphere systems. Density Functional Theory is based on the theorem that the free energy is a unique functional of the density 8. The classical version of DFT used for liquids, relies on the idea that, at the freezing transition, the correlation length is of the order of a few atomic spaces, thus mean-field theory is suitable for treating phenomena that depend on such length scales (or greater). In the context of liquid theory and crystallization, this idea implies that the thermodynamic information about a non-uniform system, such as a solid, can be found within a good approximation from the knowledge of the uniform or liquid state. Therefore, by using the knowledge of the liquid phase, given by quantities such as the pair correlation function, it is possible to approximate the thermodynamics of the solid phase. In this way, the physics contained in the liquid state is used to describe the solid phase. Density Functional Theory has been extensively studied by a number of authors 5. Subsequently, I will summarize the main points of the theory following the discussion by Oxtoby 8. We can define a functional of the density ρ() r of the system as: Ω = F[ ρ] + drρ () r V () r µ drρ() r (1) v ext 11

17 where F is the Helmholtz free energy, ρ( r ) is the density of the system, µ is the chemical potential, and V ( r ) is an external potential applied to the system. It can be ext shown that Ω v is a unique functional of the density. When evaluated at equilibrium, ρ = ρ () r, 0 Ω v gives the grand potential of the system minimized at ρ = ρ () r, i.e.: 0 Ω= PV. This functional is δ Ωv[ ρ] δρ() r ρ0 ( r ) = 0 (2) In practice, we lack a complete knowledge of the grand potential of the system. Therefore, to obtain a useful theory, approximations to the Helmholtz free energy are needed. In general, F is written as ideal term plus an excess term. The ideal term is the free energy of the ideal gas and is known exactly. However, approximations are needed for the excess term: F[ ρ] = F [ ρ] + F [ ρ] (3) ideal excess Of particular usefulness in DFT is the direct correlation function (DCF), which provides the basis for many of the approximations used in the literature 4. It is the input to the theory since it includes the response of the system to changes in the density and contains the physics of the particular problem that is being addressed. The direct correlation function of the liquid is related to the excess term of the Grand Potential: 2 δ F ρ0 r excess 1 r2 = β δρ r1 δρ r2 c[ :, ]. ( ) ( ) (4) 12

18 The direct correlation function, in general, can be obtained by using integral equations such as the Percus-Yevick closure 9, as explained later in this thesis. In some cases, DFT has been used to obtain more accurate expressions of the DCF of the liquid based on the input from Integral Equation Theories 5. II.1.2 SecondOrder Theory of Freezing In this section, I describe a theory of freezing based on the density functional formalism of non-uniform classical liquids, reviewed by Evans 10. Ramakrishnan and Yussouff formulated an early form of the density functional theory of freezing 6. Haymet and Oxtoby 11 reformulated this theory in terms of classical density functionals. The free energy functional is written as a perturbation expansion (in the density) of the excess free energy, using the uniform equilibrium liquid density as the reference system. As stated, the free energy can be naturally split into ideal and excess terms. The grand potential is then approximated by writing the excess free energy of the solid phase as a functional Taylor series about the uniform liquid, truncated, of course, to some finite order of the density. For mixtures, the Grand Potential functional is written as: ν α α ρ α α β( Ω Ω 0) = dr ρ ( r)ln ( ρ ( r) ρ0 ) α α 1 ρ = 0 ν 1 αγ α α γ γ d d C ( ) ρ () ρ0 ρ ( ) ρ0 2 r r r r r r αγ, = 1 (5) where β is the inverse of the temperature and the summation is over the number of species, ν, and the superscript of ρ specifies the species of the particles. The functions C αγ are the direct correlation functions of the liquid 4, with the superscripts referring to 13

19 various species. Although Equation (5) has been truncated at second order, the expansion can be written to any higher order of the density. However, the direct correlation functions are typically unknown above second order. Many studies have used a second-order truncation of the density functional, as we do here 5. Second order approximations consistently overestimate the coexistence density, even more so as the softness of the spheres is increased 5. Some authors have conducted studies treating higher order corrections, including the weighted density functional theories (WDFT) 12. These only require the two-particle direct correlation function despite continuing higher order term. In most cases, the same qualitative picture is found. To explore the density functional theory of freezing, the liquid direct correlation function (DCF) in addition to some knowledge of the crystalline phase is needed. Integral equations are used to calculate the direct correlation functions 4. For hard sphere systems, I use the popular Percus-Yevick (PY) scheme for which an analytic solution has been found 9. Many other integral equations provide a way of calculating the liquid DCFs. However, for anything more complex than hard spheres, solutions must be found numerically. II.1.3 Weighted Density Functionals Second-order truncation to the Taylor expansions (in density) give approximations to the excess free energy that are simple and can be solved relatively easily. Nonetheless, this approach is only an approximation. In some cases, the results obtained using them are neither qualitative nor quantitatively accurate. 14

20 A more accurate approach uses effective densities. The excess free energy is a functional of an effective density, which in turn is a functional of the real density of the liquid. The most popular and successful of these approaches is the modified weighted density approximation (MWDA), which is defined by the following set of equations 12 : F [ ρ]/ N = f( ˆ ρ) excess ˆ ρ = drdr w( r r ; ˆ ρ) ρ( r) ρ( r ) drw( r r ; ˆ ρ) = 1 (6) The kernel w( r r '; ˆ ρ), in the integral is calculated using the condition that, in the homogeneous limit, the second functional derivative of the excess free energy reduces to the direct correlation function of the liquid: 2 δ F [ 0 : 1, 2] lim excess c ρ r r = β ρ ρ0 δρ( r1) δρ( r2) (7) This equation generates a self-consistent condition that is used to calculate the effective density ˆρ. The weighted density approximation produces better results 12 than the second order truncations. In particular, Lindemann parameters improve as compared to those obtained from the Taylor expansion. Its drawback lies in the significant increase of computational effort needed to find solutions. The approximation can also easily be generalized to mixtures

21 II.2 Integral Equation Theory of Liquids One popular technique used in the study of liquids and glasses is based on integral equations 4. This approach to liquid theory was developed over twenty years ago and has been very successful in describing the liquid state for numerous systems. It has also been used to describe the super-cooled state with relative success 14. With these methods, structural and thermodynamic information can be obtained in a simple way using numerical techniques that have been improved over the years 4. These numerical techniques are relatively easy to implement and required less computational effort than usual simulation methods. The basis of this theory and how it can be used to approach the study of the glassy state is explained in this chapter. II.2.1 Pair Correlation Function The Radial Distribution Function forms the basis of the description of a liquid; it is defined for a homogeneous system as 2 : V N g( r r ) e drdr K dr (8) 2 β ( r1 Kr ) 2 1 = Z N 3 4 N where Z N is the partition function, and 1 Φ( rk r N ) is the total potential energy. The function g( r ) is proportional to the probability of finding two particles separated by a distance r1 r 2 irrespective of the position of the remaining N-1 particles and contains an 16

22 implicit average over them 4. For an isotropic system, g is a function only of the distance r r. The radial distribution function is important in the theory of liquids since it 2 1 connects thermodynamic and structural properties. II.2.2 Structural and Thermodynamics Properties The radial distribution function is needed to calculate thermodynamic properties of the liquid. The pressure and internal energy can be obtained directly from g( r ). With some additional calculations, the entropy and free energy can be calculated as well. If we assume the potential can be written as a sum of pairwise terms and that it is spherically symmetric, the pressure can be expressed as 2 : β P 2 ( ) 1 ( ) d φ r 2 βπρ g rr rdr ρ = 3 (9) dr The expression is called the Virial Equation of State, where P is the pressure, ρ the density and φ( r) is the pair potential between two particles. The energy is also related to g() r through the following relation: E N ( ) ( ) 2 = NkT + πρ g r φ r r dr (10) 17

23 From this and the following derivatives: F P = V F S = T T V (11) we are able to obtain the free energy and entropy. Thus, all the required thermodynamics quantities are accessible from the knowledge of g( r ). In practice, to get F and S, we need to perform an integral along a path defined by particular densities and temperatures, which may traverse a region of thermodynamic instability 4. Another useful thermodynamic relation is the isothermal compressibility given by: β P ρ T = ( ρktκ ) B T 1 (12) This relationship is called the Compressibility Equation of State. As we will see below, this equation is important since it places a thermodynamic constraint on g(r). Structural information on the system is obtained from the pair distribution function itself and its relation to the structure factor S( k ). The structure factor S( k) is experimentally accessible, for instance, through X-ray diffraction. The Fourier transform of g( r ) and S( k ) are related as follows 2 : S( k) = 1 + ρ ( g( r) 1)exp( ik r) dr (13) 18

24 From this quantity we can obtain information about the structure of the liquid in both real space (r) and momentum space (k). II.2.3 Integral Equation Approach How do we calculate g( r )? A very successful answer to this question has been given by The Integral Equation Theory of Liquids (IET) 4. IET has given very good results in a broad range of temperatures and densities for a wide range of systems. Two equations form the basis: the Ornstein-Zernike equation and a second one obtained from an expansion of the partition function 4 ( ) hr () = cr () + ρ c r r hr ( ) dr (14) ( ) ln g() r = βφ() r + g() r + c() r + E() r (15) where hr ( ) = gr ( ) 1 describes the fluctuation of the pair distribution function around its ideal-gas value. Equation (14) can be considered as the definition of c(r) the direct correlation function. It expresses the fact that the total correlation h(r) of the liquid is equal to a direct effect but is also dressed by the sea of particles in the liquid. The Fourier transform of the Ornstein-Zernike equation is a convolution, hk ( ) = ck ( ) + ρhkck ( ) ( ), which is used to simplify finding solutions of such integral equations. Equation (15) relates g(r), c(r) and an unknown function E(r), which is called the bridge 19

25 function. It can be formally obtained from equation (8). To get this equation we have to make an expansion in pair terms and factor out g(r), c(r), and φ ( r). The remaining terms is what we define to be E(r). At this stage no approximation have been made; these two equations are exact. Therefore with knowledge of E(r) we would be able to find the exact g(r). In practice we approximate E(r) in different ways obtaining various pairs of equations which give two equations with two unknown functions that can be easily solved, in most cases using numerical methods; thus we obtain an approximate solution for g(r). Some of these approximations will be explained in the following sections. II Hypernetted Chain Equation The first approximation is to set E(r) to zero. This is called the Hypernetted Chain Equation (HNC), which was one of the first approximations applied to many different systems. In many cases, very good results have been obtained for thermodynamic and structural properties 4. It seems to be optimal for the Lennard-Jones system 4. In the HNC approximation, g(r) is written as: g() r = exp[ βφ() r + h() r c()] r (16) II Percus-Yevick Equation A second approximation is Percus-Yevick equation 9 (PY) given by: βφ ( r) g() r = (1 e )() c r (17) 20

26 The PY equation has been used in different systems giving its best results for hard and soft spheres 2. This equation along with the Ornstein-Zernike equation can in general be solved using numerical methods such as iterative procedures. For hard spheres, the PY equation can be solved analytically. The Percus-Yevick approximation is very successful for systems that have a repulsive interaction potential such as hard spheres. It has also been applied to many other onecomponent model systems producing results with reasonable accuracy 4. Most of the calculations performed in this thesis for hard spheres systems, were done within the Percus-Yevick approximation. II Rogers and Young Equation The equations discussed previously do not possess the property called thermodynamic selfconsistency 4. This property is obtained when both expressions for the pressure, equations (9) and (12), give identical values. One approach to this problem has been given by introducing a new equation that is an interpolation of the HNC and the PY equations since the exact result given by Molecular Dynamics falls in between the results given by these two equations. The new approximation will have an extra parameter used to find the expression that possesses this thermodynamic consistency. This parameter is set using equation (12). One technique widely used to obtain thermodynamical consistency is constructed by means α of an interpolating factor: f( r; α) = 1 e r, which is called Mayer Function 2 and has been used on some equations giving excellent results. Using this function we can write an 21

27 expression for g(r) and we can chose α so that the equation is thermodynamically consistent when equation (12) is satisfied. The Rogers and Young equation is one such scheme and has proven to be a reliable approximation for obtaining g(r) for liquids. It has been used widely for different systems with remarkable results in most cases. The result of this closure is a hybrid integral equation, which interpolates between the PY and the HNC integral equations by using an appropriate weighting function as mentioned above. Solutions of the Rogers-Young equations agree well with Monte Carlo results 4. The approximation is explicitly given by the following equation: α [ ] f ( r; ) h( r) c( r) βφ ( r) e 1 gr () = e 1+ f() r (18) This approximation gives satisfactory results for hard-sphere and soft-sphere systems. Its extension to mixtures is immediate, giving accurate results for some of these systems 4. The above-mentioned equation is used throughout my research work to study soft-sphere systems. II.2.4 Generalization to Mixtures The extension of Integrals Equations to mixtures starts from the generalization of the pair correlation function gγν () r which is the probability that a particle of type γ is found between a distance r and r + dr from the origin where a particle of type ν is located. The analog expression for mixtures of equation (15) will be written as: 22

28 ( βφ γν ( r) + h γν ( r) + c γν ( r) + E γν ( r)) g () r = e (19) γν The Ornstein-Zernike equation for multi component liquids reads ( ) h () r = c () r + ρ c r r' h (') r dr ' (20) αβ αβ ν αν νβ ν The aforementioned equations, i.e. PY, HNC, RY, are easily generalized by looking at the form of the bridge function E(r) and writing the appropriate extension to mixtures. The HNC equation has a very easy generalization since E(r)=0. For Percus-Yevick, the generalization reads: βφ γν ( r) g () r = (1 e ) c () r (21) γν γν The extension of the Rogers and Young equation offers a few choices since the parameter α, in the interpolating factor, could depend on the type of the particles. The improvement from using a common factor for all the types of particles to having a different one for each type is very small in most cases, and the qualitative picture is not much improved. In all the calculations presented in this thesis, I used a single interpolating factor (identical for all species). 23

29 II.3 Hard-Sphere and Soft-Sphere Potentials In this section, I review the two potentials used through my research. A summary of its properties will be given below. II.3.1 Hard Spheres The hard-sphere potential is used in the simplest possible model of a fluid. It is generally used for systems in which the physics is mostly driven by the hard core of the particles. It has the following form: r < σ φ() r = 0 r > σ (22) where σ is the hard-sphere diameter. The hard-sphere potential has been the starting point for many integral equation approximations and has been used as a reference system for much more complex potentials 2. Rosenfeld and Ashcroft 15, using the analytical solution of the Percus-Yevick approximation, developed the modified hypernetted chain closure to study more complex potentials. The application of this closure to the Lennard-Jones potential produces very accurate results that are in very good agreement with simulations 4. The direct correlation function for hard spheres can be calculated analytically within the Percus-Yevick approximation 9. This simplifies the task of studying hard spheres systems 24

30 and producing new approximations. The analytical solution for both single component and mixtures will be used in this work as the input to the calculation of the density functional grand potential. The extension of the hard-sphere potential to binary mixtures is written as: r < σ ij σi + σ j φij () r = σij = 0 r > σ ij 2 (23) where we have taken the diameters to be strictly additive. II.3.2 Soft Spheres The soft-sphere potentials I used in this thesis are defined by the following pair interaction between particles: n σ φ() r = ε (24) n r The advantage of this potential is that the excess part of the free energy and all the quantities derived from them depend on a singe parameter, given by: 25

31 3/ n 3 ε 3/ n Γ= ( ρσ ) = ρ * T *, kt B (25) Therefore the entire phase diagram of the system will be given in term of one variable, the scaling parameter Γ. In this work, I only consider soft-sphere potentials with n=12. Due to this property of the potential, soft-sphere systems have been extensively studied by different methods such as Monte Carlo simulations, Molecular Dynamics, integral equation of liquids, and within DFT itself. There is a vast amount of literature about the thermodynamics properties of this potential, for example, the phase diagram has been mapped completely, which makes it suitable for studying new theories or new methods of solutions for some of the popular theories. 26

32 Chapter III Method of Solutions Density functional theory coupled with integral equations of liquids possess a rich degree of complexity that require the use of sophisticated techniques to find solutions that correspond to non-uniform systems, e.g. solids. In what follows, I explain the methods and algorithms we used to find these solutions. I review the gaussian and plane wave methods, which have been used extensively in the literature to solve density functionals equations. Next, I present a definition and explanation of the discrete method, used in most of my calculations and discuss advantages and disadvantages of such an approach compared to the two other approaches, namely gaussian approximations and plane wave expansions A variation of the Guillan algorithm to solve integral equations 4 is described as the method of choice to find the radial distribution function. Some techniques for improving the efficiency of this algorithm are also discussed. III.1 Method of Solution of Density Functional In this section, I describe the techniques we use to find solutions, i.e. densities that minimize the grand potential within Density Functional Theory. In the first section I present a review of the popularly used methods with some comments about their drawbacks. In the second section, I explain the discrete method, which I use to find minima of the grand potential. 27

33 III.1.1 Review of Popular Methods To minimize the free energy functional, we need to parametrize the density in an appropriate manner. Several such parametrizations have been previously used, and reviewed in detail elsewhere 5,8. I first summarize two parametrizations below, neither of which I used in my calculations; both use basis functions to describe the density. First consider the expansion of the density in plane waves. The density of the assumed crystal structure is a periodic function in space, and therefore can be expanded as a Fourier sum: ρ( r) = ρ 0 1+ µ i exp( ik r ) (26) i This method has several drawbacks. As with all the expansions it must be truncated at some order. It is possible that truncating could lead to negative densities, which not only are unphysical, but also would lead to difficulties in evaluating the ideal contribution to the grand potential. To obtain the perfect crystal solution, Lagrange multipliers could be used in the functional to constrain the number of particles. Another method, often used to represent the crystal is a sum of gaussians centered at particular positions: 3/ i ε πε Ri ρ() r = A exp r R (27) 28

34 where A is a normalization constant that can be used as a variational parameter and is unity when the crystal is perfect, i.e. one particle per site. The summation is performed in real space, over the lattice sites of the chosen crystal structure. The main drawback of this approach lies in that it restricts the shape of the density. Most calculations that use this approach set the normalization constant A equal to unity 5. III.1.2 Discrete Method As explained previously, the gaussian and plane-wave methods introduce constraints in the calculation that do not allow one to find very accurately minima of the grand potential. We define and introduce the discrete method in this section. This is a constraint free minimization method that allows one to obtain the density that is a minimum of the grand potential given the unit cell, reference density, and concentration of the system. In order to minimize the grand potential given in equation (5), we use a discrete version of the functional: ρ 1 β ρ ρ ρ ρ ρ α= 1 i ρl 2 α, γ= 1 q ν α ν α i α α αγ α γ ( Ω Ω l) = i ln ( i l ) C α q q (28) q where ρ = ρ ρ. The cubic unit cell of volume L 3 has been divided in α α α l 3 N small subcells of volume 3 a 0, with a 0 = L/ N. The grid size is defined by a 0 which determines the number of subcells within the unit cell. In each of these small subcells we have an average value for the density ρ i. The ideal free energy is calculated in real space. The excess free energy is calculated in k-space since this is computationally a more efficient 29

35 method. In addition, the hard-sphere direct correlation function is discontinuous in real space at r = σ (diameter of the particle); and numerical methods must treat this carefully. Using k-space for calculation, the excess free energy avoids this problem. The direct correlation function C αγ q is taken to be the value at the midpoint of any small cell in k- space, where q x takes the discrete values πn / L< qx < π( N 1) / L. Periodic boundary conditions are used in all our calculations. Similar numerical approaches have been used to study crystal fluid interfaces in hard-sphere and Lennard-Jones systems 16 and glassy states in hard-sphere systems 17. Figure 1. Graphical representation of the partitioning of the unit cell used in the discrete method. 30

36 For each reference density iteratively using the method of steepest descent: ρ l and each lattice constant L, we minimized the free energy α α β ( Ω Ω l ) ρ i = ρi t α ρ i (29) where ρ ' i and ρ i are the new and old values of the site-density, respectively, and the gradient is given by: ( ) α ν β Ω Ω l ρ = i ln αγ γ Cq ρqe α α ρ ρ γ = 1 q i l iqri (30) where r i is the center of cell i. The step size t is an arbitrary number chosen to be small enough, such that we obtain convergence, yet not too small to be computationally inefficient. In practice we adjust t at each iteration using the following formula, t' = t+ γ t, where γ is a variable that is increased until the gradient changes direction or the grand potential starts increasing. At this point, we set the value of γ to half the value that satisfies such conditions. I use this scheme to calculate solutions for which β ( Ω Ω l ) = 0 at its minimum, i.e. the crystal in equilibrium with the liquid. The initial condition or the starting point of the iterative process is a critical decision that affects the convergence and efficiency of the algorithm. In all cases, it was chosen to be a substitutionally disordered solid solution in a (face-centered-cubic) )fcc structure. I defined the initial density distribution to be a delta function positioned at each site of the fcc lattice. Thus, ρ at the four sites of the unit cell are set equal to 1/ ρ, where α i α l α ρ l is equal to 31

37 x α ρ, and x α α = N / Nt is the liquid concentration of particles of type α and N t the total l number of particles. Finally, ρ = N / V is the reference, or liquid density. The grand l t potential is now minimized with respect to the lattice constant L and the liquid density ρ l, until we find coexistence of the liquid and the solid phases, i.e. β ( Ω Ω l ) = 0. In this way I am able to obtain the density of the solid as well as the lattice constant. It should be noted that this is not a constrained minimization. We do not impose any relationship between the solid density and the lattice constant. The perfect crystal one without defects is not typically a minimum of the free energy. However, the periodic boundary conditions and size of the unit cell do introduce some limitations on the calculation. Since we are trying to explore freezing of an equi-concentration liquid, we start with a disordered solid solution with equal concentrations of each particle type. The underlying initial crystal symmetry is assumed and defined using discrete delta functions at the lattice sites as described previously. For the calculations performed in this work, the crystal symmetry imposed in the system by the initial condition does not seem to constrain the space of solutions to structures with similar symmetries, as is the case for the NaCl solution we found, which were obtained using an fcc structure as the initial condition. The choice of grid size a 0 is critical for our results to be meaningful. It is found that the separation between grid points a 0, should be smaller than the spread of the peaks in the density. In the gaussian method, this would be equivalent to a0 < ε where ε is defined in equation (27). By way of example, for an N=64 system with lattice constant L = 1.5σ, the grid size is a σ, and the spread of the gaussian would be around ε 0.04σ. In such a case, the grid granularity is adequate to provide accurate numerical results. As a test of the convergence algorithm, we performed a series of calculations on single component 32

38 systems, to compare with previously published results using DFT RY 18,19. We used the analytic Percus-Yevick direct correlation function for hard spheres, and the Rogers and Young closure for soft spheres. For the fcc, hard-sphere solid we find a freezing density of ρσ = Jones and Mohanty 19 find a freezing density of Their calculation had more variational freedom than previous methods. Our results are in close agreement with theirs. For the single-component soft spheres, our results are close to published results by Barrat 18. Their work finds a freezing density of ρ=1.28. Using the discrete method, the density of the liquid at coexistence was ρ l = The discrete method presented in this chapter will be used in this work to study hard spheres and soft spheres and to investigate the properties of the solutions of density functional theories. I will show that the solutions obtained using this method reveal new properties of the systems studied. III.2 Method of Solution of Integral Equations With the exception of the Percus-Yevick approximation when applied to hard-sphere systems and a few other potentials, most integral equations must be solved numerically. In general, iterative techniques are used with some suitable convergence criteria. To solve such integral equations, I initially guess cr ( ) by using the interaction potential, g( r) βu( r ) = e (31) or the solution to the integral equation in a regime where it is easier to find it. (typically in a high temperature or a low density regime). Then I used the integral equation to find the 33

39 corresponding g( r ). Then from cr ( ), hr ( ) is found using ( r) ( r) E( r) hr () = e βφ + θ + 1. θ () r = h() r c() r is calculated and from this point the iterative procedure starts: 1. Calculate g() r from the integral equation; gr () = e βφ θ ( r) + ( r) + E( r) 2. Calculate h() r from its definition: hr () = gr () 1 3. Calculate cr () from cr () = hr () θ () r 4. Calculate the Fourier transform of cr () 5. Calculate new new ( k) θ from Ornstein-Zernike eq. θ ( k) = ρc 2 ( k) ( 1 ρc( k) ) 6. Calculate Inverse Fourier transform θ new () r, from θ new ( k) 7. Mix old value of θ () r with new one θ new () r 8. Check for convergence using equation (32) 9. Go back to step 1, unless convergence condition is satisfied. These steps are repeated until the following condition is satisfied: 2 n n 1 dr 4 πr θ( r) θ( r) < ε (32) 0 where ε is normally chosen to beε 10 6, which has proven to be an excellent choice for all the application in this work. The most efficient way of performing this algorithm would utilize Fast-Fourier Transforms (FFT) to go back and forth between r-space to k-space. 34

40 The implementation of this algorithm requires that we adopt a finite grid in r-space and k- space on which the structural functions are defined. To solve the integrals associated with this theory and to perform the Fourier transforms efficiently, I have used grid sizes of 1024, 2048, 4096 points, i.e. powers of 2. The mixing of the old and new solutions in step 7 of the algorithm merit a few extra comments. There are a wide set of methods for improving the convergence of this algorithm, such as simple mixing, and all the those derived from Newton-Raphson techniques. We use a variation of Broyden s method 20 developed by Johnson 21. This algorithm has been used extensively in electronic density function calculations and it improves covergence dramatically over simple mixing techniques. As bonus, this method is memory efficient and avoids the saving of the Hessian matrix which is normally done in the traditional Broyden s method. Some integral equations, such as Rogers and Young, have an extra parameter in the bridge function E( r; α ) which is set in order to obtain thermodynamic consistency. This requires the algorithm, described above, to be part of a more general scheme that would find the parameter α. In all our calculations, a bisection method was used to determine the correct value of α, which proved to be very efficient in most cases. 35

41 Chapter IV Results and Discussion Calculations were performed on single component and binary systems of hard and soft spheres combining density functional theory with various integral equations. Results for freezing within density functional theory obtained, using the discrete method, are shown in the first section of this chapter. The solutions describe in this thesis differ from those previously published. Substantially different structures are found for some of systems I investigated. IV.1 Density Functionals In this chapter, I present the results obtained using the discrete method of solution to the DFT equations. The method is tested by comparing our results for the single component system of hard spheres and for soft spheres to previous results obtained using the gaussian parametrization and molecular dynamics simulations. I have also tested the accuracy and reliability of the method by studying its behavior as a function of the grid granularity. Results for binary hard spheres are presented in this section and compared with previous work obtained from gaussian approximation method. I also show results for binary soft spheres and compare with the solutions obtained for hard spheres. Finally, some results for ternary systems of hard spheres are presented. The results in this section illustrate that the discrete method produces more accurate results and reveals an interesting and different physics than initially found for these systems. 36

42 IV.1.1 Test of the Discrete Method Due to the inherent complexity of density functional approximations, it is important to test the implementation of any method used to minimize the grand potential. Here, I present a study of the reliability of the discrete method including a comparison with previous results. Calculations for single component systems of hard spheres and for soft spheres were performed and compared with results obtained using the gaussian method. My results are in good agreement with those found in the literature, as I will demonstrate in this section. I also studied the behavior of the grand potential as a function of the grid granularity. When the separation between adjacent points in the grid is larger than the Lindemann parameter, the results obtained are not reliable and the solutions are unphysical. If the grid size is of the order (or smaller) of the Lindemann parameter at freezing, the discrete method produces excellent results. IV Comparison for Single Component Systems I tested the discrete method by finding the freezing point of a single component hardsphere system. The results were compared with previous calculations performed by Jones and Mohanty 19. In their work they used a gaussian parametrization for the density: ρ 2 2 ( r R ) () r Ae i ε, = (27) i 37

43 and the Ramakrishnan-Yussouff grand potential was minimized with respect to both, the normalization constant A, and the width of the gaussians ε. The set of lattices positions { R i } was taken to correspond to a fcc lattice. By using A and ε as the minimization parameters, the results possess more degrees of freedom than the original work of Haymet 11. They are closer to the type of minimization that is performed when using the discrete method. By allowing the normalization constant A to change, the solutions are not constrained to be a perfect crystal structure (a crystal with one particle per lattice site). We will demonstrate that the discrete method exhibits this feature as well. Jones and Mohanty argue that this discrepancy is due to the existence of defects, which are not contained in solutions constrained to the perfect crystal. The solutions obtained in this thesis and the ones obtained by Jones and Mohanty are closer to the absolute minimum of the functional that we are investigating, since such solutions have more variational freedom giving a lower value of the grand potential. Table 1. Solid and liquid densities ( ρ s and ρ l respectively) at coexistence are shown for 3 single component hard-sphere systems (in reduced units ρσ ) Densities of hard spheres in the liquid and solid phases at freezing This work DFT-Jones Mohanty 19 MD 22 ρ l ρ s ( ρ ρ ) ρ s l l In Table 1, I show a comparison of the discrete method with Molecular Dynamics results 22 and DFT as solved by Jones and Mohanty 19. Results are expressed in reduced units of 3 ρσ, where ρ = / N V is the density of the system and σ is the diameter of the particle. 38

44 The agreement of my results with Molecular Dynamics and Jones and Mohanty is excellent. The agreement with the Lindemann parameters obtained by Jones and Mohanty is also excellent. This work gives a Lindemann parameter of 0.04 as compared to obtained by Jones. The results are also consistent with previous work, in which DFT underestimates the Lindemann parameter 5. These results demonstrate that the gaussian parametrization produces a good description for single component systems and that any asymmetry in the shape of the density has little effect on the thermodynamics of the system. A similar comparison was done for single component soft-sphere systems. The results obtained with the discrete method were compared with previous results from Barrat 18. In his paper, Barrat uses a plane wave expansion of the density to minimize the grand potential. Freezing was found at a density of the liquid of ρ=1.28. Using the discrete method, the density of the liquid at coexistence was ρ l = Thus, the discrete method is a reliable tool for studying freezing of simple liquids. Also, it provides more information about the system, such as a more detailed knowledge of the shape of the density, since the representation of the density is more robust. IV Study of the Grand Potential as Function of the Grid Size The next test of the discrete method is to study in which ways the results depend on the grid size. The results of this test are summarized in Figure 2, which shows the difference in grand potential as a function of the side of the unit cell or lattice constant L. I present results for four different values of the number of divisions (N=8, 16, 32, 64). All these 39

45 calculations were performed at a fixed liquid density ρ=0.9445, which is the freezing point predicted by Molecular Dynamics. At this density the curves for both N=32 and N=64 have common minima. These minima are located at a lattice constant value of L=1.5075, which correspond to a nearest neighbor distance of L / 2 potential in all cases satisfied the condition β ( ) Ω Ω < = The value of the grand 4 Grand Potential for various Grid sizes N=8 Grid Size N=16 Grand Potential β(ω-ω 0 ) 2 Freezing Point N= N=32 N=16 N=8 L/σ N=32 Grid Size N=64 Figure 2. Grand Potential Difference for a single component hardsphere system obtained using the discrete method is plotted. The different curves represent results for different grid sizes. Jones and Mohanty find that hard spheres freeze close to this density at a value of the liquid density of ρ= and they find a nearest neighbor distant of The Lindemann parameter is 0.048σ which is of the order of the grid size for N=32. Figure 2 also shows a second minimum for N=32 and N=64 and only one minimum for N=8 and 40

46 N=16. These minima are related they are numerical artifacts due to the grid size a. 0 A detailed look at the density profiles for these false minima reveals that the peaks at the fcc sites have no spread, i.e. the Lindemann parameters are equal to the grid separation a 0. Figure 2 also shows that the minima for N=8, and N=16 misses the freezing point obtained previously. A comparison of the grid sizes, N=8, a σ and N=16, a 0 0.8σ, with the expected Lindemann parameter for this system, which is 0.04σ, suggests that we should expect numerical problems for these values of N. To obtain accurate results for the hard spheres using Ramakrishnan-Yossouff (RYDFT) density functional theory, it is necessary to have a grid size smaller than the expected Lindemann parameter, i.e. the spread of a peak in the density must include several grid points rather than just one. A point to discuss later is the relationship of the glassy minima to the grid size. These results suggest that previous glassy states obtained using a similar discrete method, and discussed extensively by Dasgusta and Valls 23, are numerical artifacts due to their choice of grid size. In calculations that I performed using different grid sizes, glassy states only appeared for those values of grid sizes corresponding to the cases where the grid was too coarse, i.e. N=8 and N=16. For more accurate calculation using N=32 and N=64, I could not find a glassy state, which is in agreement with results obtained by Singh 5. This point will be discussed later in this chapter. IV.1.2 Freezing of Binary Hard Spheres In this section I consider binary mixtures of hard spheres. The goal is to look for structures that coexist with a liquid that has equal concentrations of the two types of particles. I will demonstrate that within the RYDFT, as the size ratio of constituents changes, I find that 41

47 the density of smaller particles obtain a multiple-peak structure. As the size ratio is decreased from unity to about σ 2 / σ 1 =0.4, the solid at freezing evolves from a substitutional disordered fcc structure to a NaCl structure with intermediate phase. IV Calculations Performed I searched for structures with a non-uniform density that coexisted with an equi-molar liquid, i.e. a liquid composed of particles with equal concentrations. The search was done for a series of diameter ratios σ 2 / σ 1 ranging in value from 0.1 to 1.0. In all cases σ 1 is set to unity and becomes the natural unit of length of the problem. Once the initial configuration has been defined, the grand potential difference was minimized using the discrete and iterative method explained in chapter two. I looked for the structure of the solid at freezing, i.e. the inhomogeneous phase coexisting with the liquid. It is important to emphasize that only a limited search for possible structures has been performed, using only the disordered fcc as initial condition and a unit cubic cell. The main limitation is the small size of the unit cell of the solid and the overall cubic symmetry. To break the cubic symmetry of the solid, one would need a larger unit cell, then the phase space of solutions grows as with any lowering of the symmetry of the solid. Using such a restricted volume limits the phase space of the search. To perform a more exhaustive search for other structures, such as AB 2 or AB 13, it would require using a larger and noncubic unit cell as mentioned before. It is possible that in our small cubic cell the solid becomes highly frustrated for the intermediate regions of size ratios suggesting the existence of other solutions, which the system cannot reach, for example a hexagonal structure. Such possibilities are left for future work. 42

48 IV Results for Binary Systems of Hard Spheres The thermodynamic values displayed in Table 2 are for the entire range of values of size ratio σ2 σ 1 studied in this work for binary mixtures of hard spheres. The values for single component systems are included for reference and correspond to σ2 σ 1=1.0, Table 2. A listing of certain thermodynamic values for hard spheres mixtures. The solid is in coexistence with an equimolar liquid. Size ratio σ σ 2 1 Lattice. Constant. L Liquid density ρ l Solid density ρ s β ( Ω Ω l ) Solid: Concentration of large particles. Total number of particles. ( ρ ρ ) S ρ l l 43

49 Solid and Liquid Densities at Coexistance Liquid Density Solid Density Density Size ratio σ 2 /σ 1 Figure 3. Liquid and Solid density for hard-sphere mixtures are plotted as function of the size ratio. The solid coexists with the equimolar liquid. The second and third columns show the values of lattice constant and liquid density where the liquid and solid coexist. I determined these points by the condition of β ( Ω Ωl ) 0 at its minimum. In most cases, the minimum was located with a precision of These results show that in the solid state at the freezing point, the concentration of large particles is larger than 50%, until the size of the ratio becomes smaller than 0.35, σ2 / σ 1 < At freezing, the density of the solid is within 20% of the coexisting liquid, and is close to the value of the single component system. This property suggests that the value of the lattice constant is driven by the large particles, which also determines the structure of the system. As shown in this table, the change in density at coexistence becomes negative in the range of size ratios between 0.8 and 0.5, where the crystal seems to be frustrated. This behavior 44

50 can be observed in Figure 3, where the densities of both the solid and the liquid are plotted as a function of the size ratio. Evidently for intermediate values, the frustrated solid possesses more entropy than the liquid. Table 3 shows several results at the freezing point of hard spheres for some diameter ratios. For each ratio, coexistence is found for the tabulated lattice constant (of the unit cell) and liquid density. It is again apparent from this table that the lattice constant remains very close to 1.14 for all these points. Therefore freezing occurs around the same lattice constant, at a value that is consistent with the freezing of the single component fluid. The size of the lattice constant is largely determined by the larger particles. The grand potential differences of the discrete solutions are also listed, and for comparison purposes, we list the minimum grand potential differences of corresponding solutions found using the gaussian approximation for the same density and lattice constant. The gaussian solution produces higher values than the solutions obtained using the discrete grid. The discrepancy increases as the size ratio increases, demonstrating the inadequacy of the gaussian approximation and the need for the variational freedom provided in the discrete case. Table 3. Comparison of results obtained using the discrete method with the gaussian approximation. The solid is in equilibrium with the liquid composed of equal amounts of both types of hard spheres. σ2 σ 1 ρ L L Discrete Grid βω ρ S Gaussian Approximation βω ε ρ S ε

51 To explain the nature of the discrepancy leading to the failure of the gaussian approximation, we first look at the physical nature of the density distribution solutions ρ() r. In Figure 4, I plot a contour of the density profile of the large particles, for a size ratio of σ2 / σ 1 = This is a plot of the density as a function of the position in the face of the cubic unit cell, i.e. ( xyz ) ρ 1,, = 0. One important feature to notice in this plot is that the densities are highly localized around the fcc lattice sites, showing a structure very similar to the one obtained using the gaussian approximation. Figure 4. Contour plot of the density of large spheres is displayed for a binary mixture of hard spheres for size ratio of The area shown is the face of a unit cell. The fcc structure is evident from the positions of the peaks in the density. 46