phase behaviour of oppositely charged nanoparticles: a study of binary nanoparticle superlattices

7/29/2019 Phase Behaviour of oppositely charged nanoparticles: A study of binary nanoparticle superlattices.

1/147

Phase Behaviour of Oppositely Charged

Nanoparticles: A study of BinaryNanoparticle Superlattices

A Thesis submitted

for the degree ofMaster of Science (Engg.)

in the Faculty of Engineering

by

Siddharth Sharma

Department of Chemical Engineering

Indian Institute of Science

Bangalore 560 012 (India)

June, 2011


2/147


3/147

i

. To my grandparents


4/147

Abstract

Binary nanoparticle superlattices or BNSLs have been a major area of research in

nanotechnology for the last decade. The conventional lithographic techniques are

limited to fabricate features only within a plane while these self-assembled struc-

tures are formed within a given volume. The potential applications of self-assembly

leading to crystalline structures include nanoelectronics, photonics and improving

the fundamental understanding of the bottom-up approach as well. The present

work deals with the crystallization of nanoparticles in a colloidal suspension lead-

ing to three dimensional crystals(BNSLs).

The goal is to generate phase diagrams for binary nanoparticle suspensions with

varying charge ratios between the two particles. Monte Carlo molecular simula-

tions in various equilibrium ensembles are the basis for the generation of equations

of state and calculation of free energies of the two coexisting phases. Coexistence

here implies the solid-fluid thermodynamic equilibrium and consequently the equiv-

alence of free energies. The calculation framework is pretty standard for pure sub-

stances but advanced techniques need to be used for mixtures.The focal point of

this work is the effect of charge asymmetry on the firmation and stability of BNSLs

from the binary suspension.

There are six phase diagrams for charge ratios of0, 0.2, 0.4, 0.6, 0.8 and1.0. The variables are the reduced pressure P and the mole fraction XA. Theproposed applications of BNSLs are generally dependent on substitutional order.

From the results, it is clear that symmetric mixtures are favorible for the formation

of BNSLs while charge asymmetry gives rise to solid solutions.


5/147


6/147

Contents

1 Introduction 1

1.1 Nanoparticle Superlattices . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Significance of Colloidal crystallization and BNSLs to Nanotech-nology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Photonics . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.2 Nanoelectronics . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.3 Magnetic Applications . . . . . . . . . . . . . . . . . . . . 4

1.2.4 Fundamental understanding of self-assembly . . . . . . . . 4

1.3 Self-Assembly:The route to Nanoparticle crystallization . . . . . . . 5

1.3.1 Equilibrium self-assembly . . . . . . . . . . . . . . . . . . 5

1.3.2 Nonequilibrium self-assembly . . . . . . . . . . . . . . . . 6

1.4 Characteristics of forces operating at the nanoscale . . . . . . . . . 7

1.4.1 The thermal barrier . . . . . . . . . . . . . . . . . . . . . . 7

1.4.2 Short range vs Long range interactions . . . . . . . . . . . 8

1.4.3 Order vs Disorder . . . . . . . . . . . . . . . . . . . . . . . 8

1.5 Van der Waals Forces . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.5.1 Dipole Dipole (Keesom) Interactions . . . . . . . . . . . . 9

1.5.2 Dipole Induced Dipole (Debye) Interactions . . . . . . . . . 10

1.5.3 London Dispersion Interactions . . . . . . . . . . . . . . . 10

1.6 Electrostatic forces and self-assembly . . . . . . . . . . . . . . . . 10

1.7 Colloidal dispersions . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.7.1 Lyophilic (or reversible) Colloids . . . . . . . . . . . . . . 12

1.7.2 Lyophobic (or irreversible) Colloids . . . . . . . . . . . . . 12

1.8 Stability of Lyophobic colloids and the Electric Double Layer . . . 13

1.9 The Primitive model . . . . . . . . . . . . . . . . . . . . . . . . . 15

iii


7/147

iv Contents

1.10 The DLVO theory and Yukawa Potential . . . . . . . . . . . . . . . 16

1.11 Entropic Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.11.1 Steric repulsion due to surface groups . . . . . . . . . . . . 20

1.11.2 Entropic Ordering . . . . . . . . . . . . . . . . . . . . . . 20

1.11.3 Depletion Forces . . . . . . . . . . . . . . . . . . . . . . . 22

1.12 Scope of the Present work . . . . . . . . . . . . . . . . . . . . . . 22

2 Monte Carlo Molecular Simulations 25

2.1 Monte Carlo Simulations . . . . . . . . . . . . . . . . . . . . . . 26

2.1.1 Importance Sampling . . . . . . . . . . . . . . . . . . . . 282.1.2 Metropolis Monte Carlo . . . . . . . . . . . . . . . . . . . 29

2.2 The isothermal-isobaric (NPT) ensemble . . . . . . . . . . . . . . 32

2.2.1 Derivation of the NPT Partition function . . . . . . . . . . 32

2.2.2 Monte Carlo moves in the NPT ensemble . . . . . . . . . . 35

2.3 The Semigrand ensemble . . . . . . . . . . . . . . . . . . . . . . . 36

2.3.1 The Grand-Canonical (,V,T) Ensemble . . . . . . . . . . 37

2.3.2 The Semigrand partition function . . . . . . . . . . . . . . 39

2.3.3 Monte Carlo moves in the Semigrand ensemble . . . . . . . 44

3 Methodology for calculating Phase diagrams 47

3.1 Calculation of Phase Equilibria . . . . . . . . . . . . . . . . . . . . 48

3.1.1 Outline of the general calculation scheme . . . . . . . . . . 48

3.1.2 Why thermodynamic integration ? . . . . . . . . . . . . . . 49

3.1.3 The coupling parameter . . . . . . . . . . . . . . . . . . . 50

3.2 Reference state free energy calculations . . . . . . . . . . . . . . . 52

3.2.1 Widoms particle insertion method . . . . . . . . . . . . . . 52

3.2.2 Free Energy of solid phases: The Frenkel-Laad method . . 55

3.3 Calculation of solid-fluid coexistence pressures for pure substances . 62

3.3.1 Solid-fluid coexistence pressure for hard spheres . . . . . . 64

3.4 Substitutional order . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.5 Calculation of solid-fluid coexistence points for Binary Nanoparti-cle suspensions: Substitionally disordered solids . . . . . . . . . . . 72

3.5.1 Legendre transforms . . . . . . . . . . . . . . . . . . . . . 73

3.5.2 Constructing the calculation framework . . . . . . . . . . . 74

3.5.3 Algorithm for generating phase diagrams: Substitutionallydisordered solids . . . . . . . . . . . . . . . . . . . . . . . 80


8/147

Contents v

3.5.4 Illustrative results for charge ratio 0.8: Q2Q1

= 0.8 . . . . . . 82

3.6 Calculation of solid-fluid coexistence points for Binary Nanoparti-cle suspensions: Substitionally ordered solids . . . . . . . . . . . . 89

3.6.1 Calculation framework . . . . . . . . . . . . . . . . . . . . 89

3.6.2 Algorithm for generating phase diagrams: Substitutionallyordered solids . . . . . . . . . . . . . . . . . . . . . . . . . 90

4 Phase Diagrams for Binary Suspensions of Nanoparticles 93

4.1 Simulation Details . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.2 Phase Diagram for Charge ratio zero; QBQA

= 0 . . . . . . . . . . . 94

4.3 Phase Diagram for Charge ratio 0.2; QBQA

= 0.2 . . . . . . . . . 974.4 Phase Diagram for Charge ratio 0.4; QB

QA= 0.4 . . . . . . . . . 100


= 0.6 . . . . . . . . . 1034.6 Phase Diagram for Charge ratio 0.8; QB

QA= 0.8 . . . . . . . . . 106


= 1.0 . . . . . . . . . 111

5 Conclusion 117

A Widoms test Particle insertion method 119

B Thermodynamic integration across chemical potential differences (s)121

References 125


9/147


10/147

List of Tables

4.1 Coexistence points for the binary nanoparticle suspension with QBQA

=0. The data is column wise presented as reduced pressure (P), theliquid phase Mole fraction (XLiqA ) and the solid solution phase molefraction (XSolA ). The error bars for the last decimal places are shownin the brackets adjoining the respective values. The points in thesecond and third columns represent the liquid and disordered F CCcoexistence lines shown in blue and green colors in Fig. 4.1. . . . . 95


=

0.2. The data is column wise presented as reduced pressure (P),the liquid phase Mole fraction (XLiqA ) and the solid solution phasemole fraction (XSolA ). The error bars for the last decimal places areshown in the brackets adjoining the respective values. The pointsin the second and third columns represent the liquid and disorderedF CC coexistence lines shown in blue and green colors in Fig. 4.2. 98


=0.4. The data is column wise presented as reduced pressure (P),the liquid phase Mole fraction (XLiqA ) and the solid solution phase

mole fraction (XSolA ). The error bars for the last decimal places areshown in the brackets adjoining the respective values. The pointsin the second and third columns represent the liquid and disorderedF CC coexistence lines shown in blue and green colors in Fig. 4.3. 101


=0.6. The data is column wise presented as reduced pressure (P),the liquid phase Mole fraction (XLiqA ) and the solid solution phasemole fraction (XSolA ). The error bars for the last decimal places areshown in the brackets adjoining the respective values. The pointsin the second and third columns represent the liquid and disorderedF CC coexistence lines shown in blue and green colors in Fig. 4.4. 104

vii


11/147

viii List of Tables


=

0.8. The data is column wise presented as reduced pressure (P),the liquid phase Mole fraction (XLiqA ) and the solid solution phasemole fraction (XSolA ). The error bars for the last decimal places areshown in the brackets adjoining the respective values. The pointsin the second and third columns represent the liquid and disorderedF CC coexistence lines shown in blue and green colors in Fig. 4.5. 107

4.6 Coexistence points for a binary nanoparticle suspension with QBQA

=0.8. The data is column wise presented as reduced pressure (P)and the liquid mole fraction (XLiqA ). The error bars for the last dec-imal places are shown in the brackets adjoining the respective val-ues. The points represent the CsCl-liquid coexistence curve shownin red color in Fig. 4.5. . . . . . . . . . . . . . . . . . . . . . . . . 108


=0.8. The corresponding CsCl-FCC line is shown in Fig. 4.5 inbrown color on the B- rich side of the phase diagram.The errorbars for the last decimal places are shown in the brackets adjoiningthe respective values. The data is column wise presented as reducedpressure (P) the disordered F CC mole fraction (XSolA ). . . . . . . 108


=0.8. The corresponding CsCl-FCC line is shown in Fig. 4.5 inbrown color on the A- rich side of the phase diagram.The error barsfor the last decimal places are shown in the brackets adjoining therespective values. The data is column wise presented as reducedpressure (P) and the disordered F CC mole fraction (XSolA ) . . . . . 1 0 9


=1.0. The data is column wise presented as reduced pressure (P),the liquid phase Mole fraction (XLiqA ) and the solid solution phasemole fraction (XSolA ). The error bars for the last decimal places areshown in the brackets adjoining the respective values. The pointsin the second and third columns represent the liquid and disorderedF CC coexistence lines shown in blue and green colors in Fig. 4.6. 112

4.10 Coexistence points for a binary nanoparticle suspension with QBQA =1.0. The corresponding liquid CsCl line is shown in Fig. 4.6 inred color.. The data is column wise presented as reduced pressure

(P) and the liquid mole fraction (XLiqA ). The error bars for the lastdecimal places are shown in the brackets adjoining the respectivevalues. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113


=1.0. The corresponding CsCl-FCC line is shown in Fig. 4.6 inbrown color on the B- rich side of the phase diagram. The datais column wise presented as reduced pressure (P) and disorderedFCC mole fraction (XSol

A

). The error bars for the last decimal placesare shown in the brackets adjoining the respective values. . . . . . 113


12/147

List of Tables ix


=

1.0. The corresponding CsCl-FCC line is shown in Fig. 4.6 inbrown color on the A- rich side of the phase diagram.. The data iscolumn wise presented as reduced pressure (P) and the disorderedFCC mole fraction (XSolA ). The error bars for the last decimal placesare shown in the brackets adjoining the respective values. . . . . . 114


13/147


14/147

List of Figures

1.1 Schematic showing the formation of the Electric Double Layer atthe surface of a colloidal particle. The two layers of charges are thecolloid surface and the envelope of counterions that surrounds it. . . 13

1.2 Schematic representaion of the Primitive model with colloid andion diameters specified. The larger yellow spheres are the sphericalcolloidal particles. The smaller red and the brown spheres representthe coions and counterions respectively. . . . . . . . . . . . . . . . 15

1.3 The Hard-core Yukawa Potential for various values of the inversescreening length . The values of the respective screening lengthsare adjacent to the plots. It is clearly visible how the potentialtends towards the Hard-sphere potential with decreasing values ofthe screening length. . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.1 Discrete Plots depicting the use of Monte Carlo method to calculatethe area of a circle.The yellow and red colors indicate the randomlygenerated points being respectively inside and outside the circle.(a)Shows the result with 1000 sample points or trials and the result is3.12400007247925 units2.(b)is done with 104 trials and the resultis 3.13159990310669

units

2.(c) is for10

5 trials and gives the result3.13987994194031 units2. And finally, (d) is for 106 trials and theresult is 3.14172792434692 units2. It can be seen as the numberof iterations increases, the answer tends more and more towards theexpected value i.e. . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.2 A Schematic showing a system ofN interacting particles in a vol-ume V. This system can exchange volume with the ideal gas reser-voir through the movable piston. . . . . . . . . . . . . . . . . . . . 33

2.3 A schematic of the identity switch move. The total number of parti-cles, volume and the particle coordinates remain the same for bothconfigurations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

xi


15/147

xii List of Figures

2.4 A Schematic showing thea system of N interacting particles in a

volume V. This system can exchange particles with the ideal gasreservoir of volume V0 V. The volume remains constant. A par-ticle when inside the subvolume V, interacts with the interparticlepotential. Otherwise, it acts as and ideal gas particle. The volumesdo not fluctuate. . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.5 A pictorial representation of the identity exchange principle througha simple argument of permutations for a set of four particles. Thereare two particles each of identities red and yellow. It is clear thatthe total number of ways to arrange them is 4!

2!2!. Extending this

argument to a system of N particles containing n different speciesgives N!

N1!N2!........Nn!. . . . . . . . . . . . . . . . . . . . . . . . . 41

3.1 A pressure vs density curve for a 256 particle system of hard spheres.The black and green point-markers represent the fluid and solidphase behaviors respectively. The density jump occurs at differ-ent pressures for both the phases resulting in a hysteresis across thePvs curves. The orange colored path shows this hysteresis. It isclear that the path for thermodynamic integration is not reversiblein case it contains a first-order phase transition. . . . . . . . . . . . 55

3.2 A schematic showing the creation of an Einstein crystal. The nanopar-ticles are tied to their respective lattice sites through harmonic springs.All the springs have the same spring constant . . . . . . . . . . . 56

3.3 A schematic of the Frenkel-Ladd method as applied for the calcu-lation of the free energy of solid states for the present work. It isimportant to note the effect that the contributions A1 and A2have on the potential energy. . . . . . . . . . . . . . . . . . . . . . 61

3.4 The pressure vs Density behavior for fluid and solid phases for asystem of 256 Hard Spheres. The black and green point-markersrepresent the fluid and solid phases. Each point is generated by aseparate NPT simulation for that particular pressure giving densityas an output. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.5 Curve fitting for the data set(Z1)P

vsP for the fluid phase. The curvefitted is the function 1.49768+0.00959033P

0.0000679955P2

0.254516 ln[P]. The points represent the data obtained through suc-cessive NPT Simulations. . . . . . . . . . . . . . . . . . . . . . . . 66

3.6 Curve fitting for the data set(Z1)P

vsP for the solid phase. Thecurve fitted is the function 1.80688+0.060833P0.00123366P2+0.0000117555P3 0.607602 ln[P]. The points represent the dataobtained through successive NPT Simulations. . . . . . . . . . . . . 67

3.7 Curves of Gibbs free energy vs. Pressure for both fluid and solidphases.The green and the orange curves represent the fluid and solidphases respectively. The intersection represents the point of equalGibbs free energies. With the temperature being the same for all theNPT simulations and the equality of chemical potentials imposed asa condition, the pressure comes out as the solution. . . . . . . . . . 68


16/147

List of Figures xiii

3.8 The cesium chloride (CsCl) structure is an example of substitu-

tionally ordered solid. Both types of nanoparticles are presentedthrough different colors. . . . . . . . . . . . . . . . . . . . . . . . 69

3.9 The copper gold (CuAu) structure is a type of binary face centeredcubic system. At a careful glance at any one of the unit cells. It isclear that two faces of each cube have one type of nanoparticle atthe center while the other four house the second species. Both typesof nanoparticles are presented through different colors. . . . . . . . 70

3.10 A face-centered solid solution configuration picked up from the

semigrand MC simulations for f1f2

= 100 for the system with charge

ratio of 0.6. The number of particles of species A(brown) is 140

while that ofB(blue) is 116. The system pressure in reduced unitsis 30. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.11 Fitted curve for ln12

vs. x1 for a binary nanoparticle suspension

with the charge ratio Q2Q1

= 0.8 where Qi is the charge on the ith

species. The fitted function is a third order polynomial of the form

m(x) = ax3 + bx2 + cx + d subject to the constraint10

m(x)dx = 0. 78

3.12 Pressure vs. Density (Pvs) behavior for binary nanoparticle sus-pension with charge ratio Q2

Q1= 0.8. The green and the black dots

represent solid solution and fluid phases respectively. Both the axesare in reduced units. . . . . . . . . . . . . . . . . . . . . . . . . . 83

3.13 Plot of Z1P

vsP for the fluid phase. The points represent the data.The function fitted in this case is 1.76905+0.0132016x0.368474Log[x].

84

3.14 Plot of Z1P

vsP for the solid solution phase. The points representthe data. The function fitted in this case is 2.30783+0.0537001x 0.000521098x2 0.787199Log[x]. . . . . . . . . . . . . . . . . . 85

3.15 The plot of F(P, ) vs Pressure for both the solid solution andthe fluid phases. The green and the orange curves represent thefluid and the solid solution phase respectively. . . . . . . . . . . . 86

3.16 The fitted curve for pressure vs mole fraction of species 1 in theliquid phase for the binary nanoparticle suspension of charge ratioof 0.8 and fugacity ratio of 104. The fitted function is 7.42836 +106.245x 626.043x2 + 3760.69x3 + 0.888557Log[x] and x1 =0.0970555. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

3.17 The fitted curve for pressure vs mole fraction of species 1 in thesolid solution phase for the binary nanoparticle suspension of chargeratio of0.8 and fugacity ratio of104. The fitted function is 351.872+2147.07x

8056.88x2 + 14471.1x3

93.2057Log[x] and x1 =

0.132434. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88


17/147

xiv List of Figures

4.1 Phase Diagram for a Binary Nanoparticle Suspension with QBQA

=

0. The blue and green curves represent the fluid and F CC solidsolution coexistence lines respectively. . . . . . . . . . . . . . . . 96


=0.2. The blue and green curves represent the fluid and F CCsolid solution coexistence lines respectively. The aezotrope is thepoint where the two curves intersect for the binary mixture. Theaezotrope pressure is P = 11.3246. . . . . . . . . . . . . . . . . . 99


=0.4. The blue and green curves represent the fluid and F CCsolid solution coexistence lines respectively.The aezotrope is the

point where the two curves intersect for the binary mixture. Theaezotrope pressure is P = 11.4074. . . . . . . . . . . . . . . . . . 102


=0.6. The blue and green curves represent the fluid and F CCsolidsolution coexistence lines respectively.The coexistence pressure forthe aezotropic mixture is P = 10.9727. . . . . . . . . . . . . . . . 105


=0.8. The blue and green curves represent the fluid and solid so-lution coexistence lines respectively.The red curve represents theCsCl-liquid equilibrium. The triple points are at the intersection ofthe CsCl-Liquid and the liquid-solid solution equilibrium curves on

either side ofXA = 0.5. The solid-solid equilibrium lines shown inbrown emanate from the respective triple points. . . . . . . . . . . 110


=1.0. The blue and green curves represent the fluid and solid so-lution coexistence lines respectively.The red curve represents theCsCl-liquid equilibrium. The triple points are at the intersection ofthe CsCl-Liquid and the liquid-solid solution equilibrium curves oneither side of XA = 0.5. The solid-solid equilibrium lines shownin brown emanate from the respective triple points. The curve issymmetric about XA = 0.5. . . . . . . . . . . . . . . . . . . . . . 115


18/147

Chapter 1

Introduction

1.1 Nanoparticle Superlattices

Particles of sizes ranging between 10s to 100s of nanometers can be regarded as

nanoparticles. These particles have some peculiar and interesting properties which

in most cases are different from the bulk properties of their constituent materials .

Moreover, they can self assemble to form a range of three dimensional crystalline

structures having simple to complex geometries. Such substitutionally ordered crys-

talline structures, when made up of two types of nanoparticles are commonly known

as Binary Nanoparticle Superlattices (BNSLs)(Shevchenko et al., 2005). BNSLs

have been an active area of research for chemical engineers, applied physicists,

chemists and material scientists more so in the last decade than ever before. This

can be attributed to the advances in synthesis techniques as well as the renewed

demands that result from advances in Nanoscale science and technology. As far

as classification is concerned, it would be an unfair simplification to put these ma-

terials in the same category as ionic and molecular solids as their formation pro-

cesses involve interspecies interactions that are entirely different from a chemical

bond. These interactions are commonly known as noncovalent interactions. There

are several experimental works(Shevchenko et al., 2006; Kalsin et al., 2006; Chen

1


19/147

2 Introduction

et al., 2007; Parket al., 2008; Murray et al., 2000) in the last decade that have suc-

cessfully synthesised BNSLs and have been able to produce a variety of complex

geometries . The reasons for interest in the formation and stability of BNSLs are

their prospective applications in various fields of nanoscale science and technology,

which are briefly discussed as follows.

1.2 Significance of Colloidal crystallization and BNSLs to Nan-

otechnology

The major areas of application of these structures are based on their unique proper-

ties(Murray et al., 2000). They can be broadly classified as follows.

1.2.1 Photonics

A photonic crystal can be used to control the flow of light waves just as a semi-

conductor is used to manipulate the flow of electrons. What is important here is

that most of todays computing technology is based on semiconductors and hence

the control achievable over the flow of electrons. This means that the speed of

computation is also limited by the speed at which electrons travel within the semi-

conductor crystal. If light is used instead, we could see the age ofPhotonic com-

puting. Colloidal crystals of diamond symmetry can be used create photonic crys-

tals with band-gap within the visible range(van Blaaderen, 2006). There are a few

instances(Kalsin et al., 2006; Shevchenko et al., 2006; Chen et al., 2007) where

experimentalists have been able to create colloidal crystals made up of two types

of particles. The crystalline structure of these crystals is similar to that of diamond

with each particle of one type surrounded by four particles of other type in a tetra-

hedral geometry.


20/147

Chapter 1 3

As far as photonic crystals with band-gap within the visible range are concerned,

the MgCu2 type Laves phase(Hynninen et al., 2007) provides a viable solution. In

these types of crystalline structures, the particles larger in size(Mg) give the dia-

mond geometry and the particles smaller in size(Cu) give pyrochlore geometry .

Crystals with both these geometries have band-gap in the visible range. One can

obtain either the diamond structure or the pyrochlore structure through selective re-

moval of one of the particle types by chemical or thermal processes.

The quest for a desired geometry has now led to what is known as programmable

self-assembly. For example, complimentary DNA strands as patches on colloidal

particles could provide a predetermined structure due to the selectivity of the strands

i.e. adenine with thymine and guanine with cytosine(Park et al., 2008). Such

patches when tetrahedrally arranged could result in a diamond like lattice.

1.2.2 Nanoelectronics

All types of lithographic techniques can only fabricate nanostructures in a sin-

gle plane. Nanoparticle self-assembly can potentially be used to fabricate nanos-

tructures within a given volume. It could be the next step through which a three

dimensional network of transistors can be achieved such that the nanocrystal con-

sists of an insulating and conducting material with the possibility of addressing each

unit individually(van Blaaderen, 2006). Pertaining to semiconductors, since each

semiconductor nanoparticle is a quantum dot, therefore a binary superlattice could

achieve a tailored band-gap which is different from that of either of the species.

This would result in a desired band-gap matching(Chen et al., 2007). Also in some

nanocrystals, the electron transport is seen to be temperature (Doty et al., 2001) and

size distribution dependent(Beverly et al., 2002).


21/147

4 Introduction

1.2.3 Magnetic Applications

The combination of two different magnetic materials could lead to the material

with a high magnetic energy density(Zeng et al., 2002). For this to happen, one of

the materials should have high coercivity and the other should have a high magnetic

moment.

1.2.4 Fundamental understanding of self-assembly

Self-assembly as a process is not yet fully understood. To study the impact

of various influencing factors on the process in its entirety is quite difficult if one

restricts themselves to molecular self-assembly (Whitesides and Boncheva, 2002).

The reason being it is very difficult to change anything about a molecule. It is either

what it exactly is or it is not the same molecule at all. For instance, there is nothing

like strongly or weakly charged acetate group. The charge on an acetate group is

always minus one. Experimentalists involved in synthesis can vary the structure of

a molecule or the charge it will carry to a very large extent as a result of sophistica-

tions in synthetic chemistry but once a species is formed, there is very little that can

be done.

Nanoparticles, or for that matter colloidal particles, provide that much required

room for variation in properties like size, charge, shape etc. This presents an oppor-

tunity to study the variation in the properties of the final structure with that of the

building blocks.


22/147

Chapter 1 5

1.3 Self-Assembly:The route to Nanoparticle crystallization

There is no complete agreement on the definition of self asembly. The defini-

tion given in a paper by George.M.Whitesides (Whitesides and Boncheva, 2002)

is widely accepted and is as follows:

Self-assembly is a process in which components, either separate or linked, spon-

taneously form ordered aggregates

Self-assembly occurs at various length scales both under equilibrium and far

from equilibrium conditions. From molecules to nanoscale objects i.e. nanoparti-

cles and nanowires etc. to mesoscopic length scales involving objects that are sev-

eral micrometers or millimeters in size e.g. latex particles (Hachisu and Yoshimura,

1980; Yoshimura and Hachisu, 1983). Self-assembly is intrinsic to life processes.

The formation of folded proteins, cell membranes etc. are all examples of self-

assembly. However, almost all self-assembling biological systems are out of equi-

librium systems and the are far less understood as compared equilibrium self-

assembling systems.

1.3.1 Equilibrium self-assembly

The formation of BNSL from a suspension of nanoparticles is an example of

equilibrium self-assembly. The final tructure is an equilibrium structure. The con-

ditions for forming a crystalline structure from a suspension are dictated equilibria.

The equilibria denotes the states at which the following conditions are satisfied,

namely,

1. Tf = Ts ; The temperatures of the two phases are equal.


23/147

6 Introduction

2. Pf = Ps; The pressures of the two phases are equal.

3. f = s; The chemical potentials of the components in the two phases are

equal.

Where the subscripts f and s refer to the fluid and solid phases respectively.

The thermodynamic notation is standard. Reversibility is the most significant fea-

ture of equilibrium self-assembly (Lindsey, 1991; Whitesides and Boncheva, 2002).

It is only due to reversibility that equilibrium self-assembly leads to ordered struc-

tures such as BNSLs instead of glasses which are generally resultants of irreversible

interactions among the components that act as building blocks.

1.3.2 Nonequilibrium self-assembly

This is also known as dynamic self assembly. These type of processes lead to

ordered structures when the system is far from equilibrium. The final structure ob-

tained is usually characterized by constant intake and dissipation of energy. It is the

complexity of these processes that makes them difficult to control and hence imple-

ment. For this reason, the applications of self-assembly in biological systems arelimited. In context of the literature available for nanoparticle crystallization, drying

mediated self-assembly of nanoparticles (Rabani et al., 2003) is a nonequilibrium

process.

The present work is based on equilibrium self-assembly. The final structure in

such a scenario is the resultant of a complex interplay among various forces that

act at the length scale of the individual components, i.e. at the nanoscale (109m)


24/147

Chapter 1 7

since we are concerned with nanoparticles as the building blocks. The subsequent

sections addresses these nanoscale forces in some detail.

1.4 Characteristics of forces operating at the nanoscale

In the context of self-assembly, the issue of nanoscale forces has become more

significant in the last decade or so than ever before. The advanced synthesis tech-

niques that have come up in recent years can provide nanometer size objects i.e.

nanoparticles, nanowires etc. as building blocks for self-assembled structures such

as BNSLs. With the decrease in length scales of the individual building blocks

achieved by the synthesis of nanoparticles of sizes ranging from 10s to 100s of

nanometers, the forces that are active at those length scales also became relevant.

Before discussing various types of forces that operate at the nanoscale, it is also

important to discuss some qualitative aspects that are general to all of them (Bishop

et al., 2009).

1.4.1 The thermal barrier

The net interaction among constituent entities must be at least a few times kBT,

where kB is the Boltzmann constant and T is the temperature. Without this condi-

tion satisfied, self-assembly is not feasible as thermal motion is enough to disrupt

it. Although, if higher volume fractions are achieved, entropic effects may lead to a

self assembled structure.


25/147

8 Introduction

1.4.2 Short range vs Long range interactions

A short range interaction (Frenkel and Smit, 2002) means that the total potential

energy of a given particle i is dominated by interactions with neighboring particles

that are closer than a cutoff distance rcut. It essentially means that the potential dies

down at large distances although it is not rigorously zero for r rcut. For example,the Lennard-Jones potential

V(r) = 4 r 12 r 6 (1.1)where is the depth of the potential well, ris the interparticle separation and is the

particle diameter, is a short range interaction, whereas, the electrostatic or coulomb

potential that acts between two charged bodies is a long range interaction since it

decays as1r

.

For self-assembly, a short range interaction like LJ needs to be stronger ( higher

values of ) to compensate for the decrease in entropy associated with forming an

aggregate as it results in a loss of translational, or in some cases, rotational degrees

of freedom . A long range interaction can lead to self-assembly a lot more easily as

it can overcome the entropic effects.

1.4.3 Order vs Disorder

An equilibrium self-assembly process leads to a final structure as a result of

achieving a free energy minimum. But just as chemical equilibrium does not mean

kinetic arrest, thermodynamic equilibrium by no means implies dynamic arrest. But

as the process of self assembly goes on, there are always dynamically arrested states

like glasses or gels which compete with the structure with the minimum free energy.

Systems in which short range interactions are predominant, are much more vulner-

able to dynamic arrest and hence the formation of glassy states (Dawson, 2002),


26/147

Chapter 1 9

which are the signatures of non-ergodic systems (Lebowitz and Penrose, 1973). It

is therefore always advisable to use long range potentials for self assembly (Foffi

et al., 2002). From a different perspective, if the building blocks that self-assemble

to form the final structure are small enough that an otherwise short range interaction

is significantly strong at distances much larger than the size of the building blocks,

then even such a short range interaction is good enough to provide an ordered struc-

ture. This makes nanoparticles excellent candidates for self assembly.

In the subsequent sections, various kinds of forces that drive self-assembly are

introduced and briefly discussed.

1.5 Van der Waals Forces

In any material, be it atoms, molecules or nanoscale objects that are made up of sev-

eral atoms or molecules, charges are always on the move. This constant movement

of charges is bound to cause electromagnetic fluctuations. Van der Waals forces

is the name given to the forces arising out of these fluctuations. With proper tun-

ing, vdW forces are quite useful for nanoparticle self-assembly (Harfenist et al.,

1996). These are generally attractive forces and can be broadly classified into three

categories(Bishop et al., 2009; Hunter et al., 1989; Israelachvili, 1992).

1.5.1 Dipole Dipole (Keesom) Interactions

This type of interaction occurs between components that have a permanent elec-

tric dipole moment. The strength of the interaction depends upon the distance be-

tween the two dipoles as well as their relative orientation.


27/147

10 Introduction

1.5.2 Dipole Induced Dipole (Debye) Interactions

A component with permanent dipole moment may induce a dipole moment into

another component. This could give rise to an interaction between the two.

1.5.3 London Dispersion Interactions

These interactions arise due to incessant movement of charges within the com-

ponents that otherwise do not possess any polarity. Instantaneous dipoles are formed

all over the system and give rise to these instantaneous forces that are transient in

nature.

Van der Waals forces are short range forces decaying as r6.

1.6 Electrostatic forces and self-assembly

Unlike vdW forces that are always attractive, electrostatic forces can be attractive

or repulsive depending upon wether they are between unlike or like charged bodies.

There are two aspects to their importance in context of the present work.

They are an important factor in the formation of colloidal crystals which include

BNSLs. The present work shows how the final crystalline geometry for charged

nanoparticles differs from that of uncharged ones i.e hard spheres that can only

crystallize into f cc or hcp structures. This is an indication of the significance of

Coulomb interactions in self-assembly. The advantage with electrostatic forces as

interparticle potentials is that their magnitude and length scale can be controlled by

proper choice of solvent and the concentrations of ions surrounding the particles.

This occurs through control on the screening of electric charges on the nanoparti-

cles and of course controlling the magnitude of the charges themselves. Advanced


28/147

Chapter 1 11

synthesis techniques are capable of achieving the required control on the properties

of electrostatic forces.

Electrostatic forces prevent coagulation in colloidal suspensions by introducing

repulsion among the particles. This makes them the central factor in stability of

nanoparticle suspensions.

1.7 Colloidal dispersions

In a normal solution, the size of the solute particles is the same as those of the sol-

vent and it is assumed that the solute particles are uniformly dispersed within the

solution. The solute particles are in motion at all times due to the kinetic energy

they posses as a result of the finite temperature of the solution. It can therefore

be reasonable to assume that solvent molecules are kinetic units that are uniformly

dispersed within the solvent giving rise to the thermodynamic system commonly

known as a solution. However, in some cases the kinetic units that are dispersed

within the solvent are much larger than the solvent molecules. Such systems are

known as colloidal dispersions (Hunter et al., 1989).

In general, if a substance A is insoluble in a substance B, then it is always pos-

sible to break substance A into smaller and smaller pieces such that it is almost

uniformly distributed in substance B. In such a case, substance A is known as the

dispersed phase and substance B is called the dispersion medium. There is a set of

terminology (Hunter et al., 1989) that classifies colloidal dispersions on the basis

of the states of matter the dispersed phase and the dispersion medium, e.g., gels,

aerosols etc. The present work deals with two species of nanoparticles (dispersed

phases) suspended within a solvent (dispersion medium). Such a system is called

a Binary nanoparticle suspension. There are no sharp size limits as far as sizes of


29/147

12 Introduction

colloidal particles are concerned. The lower size limit is said to be 1nm as par-

ticles smaller than this would eventually form a solution rather than a colloidal

dispersion. The upper limit is somewhat close to 1m although particles larger than

that do show colloidal behavior. The classification that is much more significant

is based on the affinity of the dispersed phase towards the dispersion medium or

equivalently on the spontaneity of dispersion if the solvent is added to a dry colloid.

In that framework, there are two types of colloids.

1.7.1 Lyophilic (or reversible) Colloids

These are characterized by a decrease in the Gibbs free energy as the dispersion

medium is introduced into the dry solute and hence are thermodynamically sta-

ble (Hunter et al., 1989). There is strong interaction between the dispersed phase

and the dispersion medium which provides enough energy to break up the disperse

phase. An increase in entropy is also there. Any decrease in the solvent entropy is

generally overcompensated by the increase in the entropy of the solute. It is the

spontaneity (G0) of the process that would allow the formation of a colloidal dis-persion even after the separation of the dispersed phase from the dispersion medium

and therefore the name reversible colloids.

1.7.2 Lyophobic (or irreversible) Colloids

In this case there is an increase in the Gibbs free energy as the dispersion

medium is introduced into the dry solute and it is minimum only when the dis-

persed phase exists as a single coagulated lump (Israelachvili, 1992).Therefore,

such systems can only remain in colloidal state if there is a strong repulsion among

the particles of the dispersed phase. This way coagulation can be prevented for long

periods. It is important to keep in mind that the system is still thermodynamically


30/147

Chapter 1 13

Figure 1.1: Schematic showing the formation of the Electric Double Layer at the surface of

a colloidal particle. The two layers of charges are the colloid surface and the envelope of

counterions that surrounds it.

unstable and will form an aggregate over longer periods of time. The present work

pertains to nanoparticle suspensions as lyophobic colloidal systems that are stable.

1.8 Stability of Lyophobic colloids and the Electric Double Layer

There are predominantly two methods to stabilize lyophobic colloidal dispersions.

1. Electrostatic stabilization

The particles of the dispersed phase can be given a charge ( positive or negative ).


31/147

14 Introduction

If all of them have the same charge, they will repel each other and this will prevent

coagulation. This type of dispersions are called charged stabilized.

2. Steric stabilization

The particles can be coated with a material that itself prevents close approach that

could lead to coagulation. These type of systems are known as steric stabilized. The

present work considers only charged stabilized colloids. The mechanism of charge

stabilization is not straightforward and the rest of the discussion on electrostatic

forces deals with the same in detail.

In case of nanoparticles it is generally the capping ligands ( surfactant coating

) that carry the charge and not the nanoparticle itself. The ions that are released

from the surface groups are known as counterions as they have charges opposite to

the charge on the colloid. The other kind of ions are the coions that arise from the

dissociation of the solvent. It is important to consider that the process of solvent

dissociation also gives counterions. The total excess amount of coions and coun-

terions is known as excess salt. The counterions are always in thermal motion but

they are also attracted to the colloid surface. As a result they build up a layer of

charge opposite to that of the colloid itself and hence there is the creation of what

is known as the Electric Double Layer (Fig. 1.1). The name is apt as the first layer

is the charge on the surface group of the colloid and the second layer of opposite

charges is formed by the accumulation of counterions around the colloid surface.

Proper description of charged colloids requires accurate modeling of the their be-

havior in the dispersion. The simplest model of spherical charged colloids is the

primitive model and is based on the coulomb potential. More sophisticated models

are derivatives of the primitive model with approximations regarding the ion con-

centrations.


32/147

Chapter 1 15

Figure 1.2: Schematic representaion of the Primitive model with colloid and ion diameters

specified. The larger yellow spheres are the spherical colloidal particles. The smaller red

and the brown spheres represent the coions and counterions respectively.

1.9 The Primitive model

According to the primitive model (Belloni, 1986; Hynninen, 2005) the colloidal

particles are spheres with a certain amount of charge whereas the coions and coun-

terions are spheres of a different diameter ( smaller than the colloidal particles )

carrying the elementary electronic charge with the same and opposite sign from the

colloid respectively (Fig. 1.2). For example, if the colloid is a sphere with charge

-Ze and diameter , then the coions and counterions are spheres of diameter ion

with charges -e and e respectively. The solvent in this case is a continuum with

dielectric constant S.


33/147

16 Introduction

The interaction V(r) among particles is given by Coulombs law

V(r)

kBT=

ZiZjBr

for r ij for r < ij

(1.2)

where B=e2

40SkBTin S.I units, is the Bjerrum length which is the length at

which the electrostatic interaction becomes comparable to the thermal energy scale

kBT . Zi and Zj are the charges on the particles. Both i and j could denote either

colloid or any of the ions. For colloid-colloid interaction ij= , for colloid-ion

interaction ij=(+ion)

2and for ion-ion interaction ij= ion.

The primitive model is a very crude simplification and has some limitations. The

most significant one is that simulations involving the primitive model become com-

putationally demanding for colloids that have high surface charges as the number of

counterions that have to be included to neutralize the colloidal charge rises steeply.

It is quite clear that coarse-graining with respect to the coion and counterion popu-

lation is needed to have a more usable model for charged colloids. The most widely

accepted model is the DLVO theory.

1.10 The DLVO theory and Yukawa Potential

The theory was proposed by Derjaguin and Landau (Derjaguin and Landau, 1941),

and separately by Verwey and Overbeek (Verwey et al., 1948). Hence the name

DLVO theory. It addresses one of the main shortcomings of the primitive model by

considering the co- and counterions as point like charges with no dimensions. This

is clearly a convenient simplification but as it will be shown, it works very well. The

other basic considerations are on the lines of the primitive model. We consider a

spherical colloid with charge Ze and diameter suspended in a continuum solventwith dielectric constant S. The temperature of the system is T. Let the density of


34/147

Chapter 1 17

ions within the solutions at a large distances from the colloidal particles be 2sol

or otherwise the bulk salt concentration issol. The DLVO theory proposes that the

coion and counterion densities can be represented by the Boltzmann distribution.

Therefore the coion density is given by

(r) = sol exp [(r)] (1.3)

And the counterion density is given as

+(r) = sol exp[(r)] (1.4)

where (r) = e(r)kBT

, and (r) is the electric potential from the center of the

colloidal particle. The local charge density would be the difference between the

coion and counterion density and is therefore esol(exp [(r)] exp[(r)]) or2esol sinh( ekBT). Substituting this into the Poisson equation gives the nonlin-ear Poisson-Boltzmann(PB) equation (Gouy, 1910; Chapman, 1913; Debye and

Huckel, 1923). i.e.

2(r) = 2 sinh[(r)] (1.5)

where =

8Bsol in S.I units, is the inverse Debye screening length. The

boundary conditions invoked are the potential is zero at infinite distances and the

field strength has a finite value at the surface of the sphere. So, they are written as

(r) = 0 for r (1.6)

|n (r)| = 4BZ2

for r =

2(1.7)

where n is the unit normal vector to the surface of the spherical colloid. For solving


35/147

18 Introduction

analytically for a spherical geometry, the PB equation needs to be linearized as

2(r) = 2[(r)] (1.8)

Equation (1.8) with boundary conditions (1.6) and (1.7) gives the solution as

(r) = ZB exp(2

)exp(r)(1 +

2) r

(1.9)

The pair potential between two charged colloids can found out from (1.9). The

assumption is that the electric double layer of the two particles do not disturb

each other. The resultant pair potential is known as the Screened Coulomb or

the Yukawa Potential.

V(r)

kBT=

ZiZjB(1+

2)2

exp[(r)]r

for r

for r <

(1.10)

The second part of this piecewise defined function denotes the hard core nature

of the potential. To be complete in our description, this potential is the Hard-core

Yukawa Potential. This prevents the overlapping of particles during simulation.

In context of the present work the simulations are done in reduced units. For that

purpose, the length is scaled by the particle diameter and the energy is scaled with

the contact energy. So in this case the potential is written as

V(r) =

ZiZj

exp[(1r)]r

for r 1 for r < 1

(1.11)

The dependence of the Yukawa potential on the screening length is shown in

Fig. 1.3. It is assumed that the refractive index of the colloidal particles and the

solvent in which they are dispersed is the same. This is a requirement to neglect the

London dispersion vdW forces (Hynninen, 2005).


36/147

Chapter 1 19

Figure 1.3: The Hard-core Yukawa Potential for various values of the inverse screening

length . The values of the respective screening lengths are adjacent to the plots. It is

clearly visible how the potential tends towards the Hard-sphere potential with decreasing

values of the screening length.


37/147

20 Introduction

1.11 Entropic Forces

It is a well known fact that hard spheres crystallize into face-centered cubic and

hexagonal close-packed geometries upon sufficient increase in volume fraction (Pusey

and Van Megen, 1986). So, self-assembly can occour even without the presence of

an attractive potential. This emphasises the importance of entropic effects. Also in

some cases the interparticle interaction is too strong to form an ordered structure

and entropic effects offer the required weak repulsion (Bishop et al., 2009). There

are mainly three ways in which entropic effects can influence nanoparticle crystal-

lization.

1.11.1 Steric repulsion due to surface groups

Nanoparticles are generally charge stabilized through capped ligands dissoci-ating and acquiring a net charge at the particle surface. If the attractive potential

becomes too strong then it can be controlled by providing a steric repulsion. This

is done by choosing long chain molecules as surfactants so that each nanoparticle is

surrounded by a polymer brush (Milner, 1991; Currie et al., 2003). The neighbor-

ing polymer brushes repel each other and provide a relatively weak steric repulsion.

The term relatively weak here means that the entropic repulsion is enough to pre-

vent the formation of a disordered coagulated lump due to the presence of a strong

long range electrostatic attraction but is not so strong that it actually prevents the

formation of the Nanoparticle superlattice (BNSL).

1.11.2 Entropic Ordering

Consider a concentrated system of nanoparticles under constant pressure and

temperature that interact as hard spheres. This means their potential energy is zero


38/147

Chapter 1 21

when they do not overlap and infinite otherwise. The Helmholtz free energy for a

system is defined as

A = K.E T S (1.12)

Where K.E is the kinetic energy of the system. Now the phase transition occurs

because the Helmholtz free energy of the BNSL is less than the fluid phase.This

gives

K.E T Sf > K.E T Ss (1.13)

where the subscripts f and s stand for fluid and solid phases respectively. This

leads us to the result

Ss > Sf (1.14)

which means that the entropy of the solid phase is more than that of the fluid phase .The thermodynamic explanation can be supplemented by the following argument.

In case of Hard Spheres, entropy is proportional to the available free volume.

As the volume fraction of the particles increases in the system, there comes a state

where the available free volume per particle diminishes such that they cannot move

around. In such a scenario they undergo a phase transition and go into a solid state

which is the only way they can increase the available free volume. The reason for

the phase transition is the increase in the availability of the local free volume for a

nanoparticle to traverse and perform the translational and vibrational motions about

the lattice site. In other words, the system has sacrificed macroscopic disorder to

gain microscopic disorder (Eldrldge et al., 1993; Chen and OBrien, 2008; Cottin

and Monson, 1995). This is just another way to look at Eq. (1.14).


39/147

22 Introduction

1.11.3 Depletion Forces

These type of interactions are examples of attractive entropic interactions. They

are active when two colloidal particles are so close to each other that even the small

solvent molecules cannot access the space between their surfaces. This leads to an

osmotic pressure that pushes the particles towards each other. This results in a com-

mon excluded volume which means that the net excluded volume is lesser than what

it was before when the particles were far apart. This means more volume available

to the solvent molecules and hence an increase in entropy consequently leading to

a decrease in free energy (Asakura and Oosawa, 1958).

After discussing the forces operating at the nanoscale, the next step is to imple-

ment the interparticle potential into a simulation methodology and to calculate the

relevant properties.

1.12 Scope of the Present work

The objective of this work is to study the phase behavior of oppositely charged

nanoparticles. Charge asymmetry is one the central areas of investigation in col-

loidal crystallization and that has been our motivation. The focal point of study is

the variation in charge ratio of both the species and its effect on the phase diagrams.

This is accomplished by calculating phase diagrams and studying the stability of

BNSLs. For such a calculation, solid-fluid coexistence points need to be calculated.

This is done through calculating the free energy of both the phases and thereby ob-

taining the values of the phase variables at the point where the free energies of both

phases are equal. For a single component, first the Gibbs free energy needs to be

calculated for initial states i.e the fluid and the solid state.


40/147

Chapter 1 23

The calculation for a binary mixture is more involved. For one, it involves MC

simulation in the semi-grand ensemble which allows identity switch MC moves

between the two species and thereby allows the existence of Substitutionally disor-

dered solids or solid solutions . Also, the thermodynamic integration needs to be

done across two variables, pressure and fugacity ratio in this case, as compared to

just pressure in case of a single component.


41/147


42/147

Chapter 2

Monte Carlo Molecular Simulations

This chapter includes a description of molecular simulation methods using Monte

Carlo technique.The simulations for the present work were carried out in the canoni-

cal(NVT), isobaric-isothermal(NPT) and the semigrand ensembles. The description

is kept brief. For details the reader should refer to (Frenkel and Smit, 2002; Allen

and Tildesley, 1989). First, the Monte carlo method is discussed with an introduc-

tion to concepts like importance sampling. Thereafter, the Metropolis MC scheme

is explained for the canonical ensemble. The canonical ensemble is quite standard

and the partition function for the same will be the starting point of the discussions

pertaining to the NPT and semigrand ensembles. The discussion on the Metropolis

algorithm is based on the NVT partition function.

The present work is based on molecular simulations and thereafter numerical

techniques to calculate the Phase diagrams for oppositely charged nanoparticles in

suspensions. The details of the methodology will be discussed in the next chapter.

A brief introduction to Monte-Carlo simulations is presented here.

25


43/147

26 Monte Carlo Molecular Simulations

2.1 Monte Carlo Simulations

The Monte Carlo method is basically a way to solve integrals. It does so through the

generation of random numbers. A simple example of using a Monte Carlo method is

the calculation of the area of a circle of radius 1 unit in a certain coordinate system .

This can also be calculated analytically as r2 where r is the radius, thereby giving

the answer as . This is done through solving the integral

4 10

(1 x2dx (2.1)

The factor of four comes as the integral computes the area of the circle lying in

the first quadrant only. Using the Monte carlo method for this calculation involves

generating random numbers between the coordinate values of (-1,-1) and (1,1) i.e.

within the square of side 2 units centered at the origin. Out of these random coor-dinates, the ones that satisfy the condition

(xi 0)2 + (yi 0)2 would lie inside

the circle of radius 1 centered at the origin. If these points are divided by the total

number of points then it is equivalent to the ratio of the area of the circle and the

square provided there are enough points to sample. The conclusion is

Area of the circle = [Number of points inside the circle

Total number of points](Area of the square)

(2.2)The same could be accomplished by generating the random numbers in a square of

side 1 with vertices as (0, 0), (0, 1), (1, 0) and (1, 1) and selecting the random points

lying within the quarter circle. Effectively, the integral in (2.1) is solved through the

Monte Carlo method.

Based on this method the area of the circle is calculated four times for different

number of points sampled. The results are shown in Fig. 2.1. The difference in

the results can be explained as follows. Say one needs to calculate the following


44/147

Chapter 2 27

Figure 2.1: Discrete Plots depicting the use of Monte Carlo method to calculate the area

of a circle.The yellow and red colors indicate the randomly generated points being re-

spectively inside and outside the circle.(a) Shows the result with 1000 sample points or

trials and the result is 3.12400007247925 units2

.(b)is done with 104

trials and the resultis 3.13159990310669 units2.(c) is for 105 trials and gives the result 3.13987994194031units2. And finally, (d) is for 106 trials and the result is 3.14172792434692 units2. It canbe seen as the number of iterations increases, the answer tends more and more towards the

expected value i.e. .


45/147


one-dimensional integral numerically.

I =

ba

f(x)dx (2.3)

Instead of conventional quadrature techniques where f(x) is evaluated at predeter-

mined values of x, one can use averaging to compute the value of I. For this the

integral is written as

I = (b a)f(x)dx (2.4)

where f(x) denotes the unweighted average of f(x) over the interval [a, b]. Thevalue of I is calculated by determining the value of f(x) over a large number of

x values, say N within [a, b]. Clearly the larger the value of N, the better is the

estimate off(x) and hence I with N giving the correct value.

Therefore, as in case of any numerical technique, the number of iterations in a

Monte Carlo (MC) method increase the accuracy of the result. The next important

question is, how should one distribute the sampling across the interval?

2.1.1 Importance Sampling

Suppose that the function f(x) in Eq.(2.4) follows a non-negative probability

distribution g(x). Then f(x) should be sampled non-uniformly across the interval

[a, b] according to g(x). For convenience let a = 0 and b = 1. Now, Eq.(2.4) can be

rewritten as

I =

10

f(x)g(x)

g(x)dx (2.5)

Now suppose g(x) is the derivative of another non-negative and nondecreasing

function h(x) such that h(0) = 0 and h(1) = 1 , i.e. g(x) is normalized. The

integral I can be written as


46/147

Chapter 2 29

I =10

f[x(h)]g[x(h)]

dh (2.6)

If h is the integration variable, then x has to be expressed as a function of h.

For, M random values ofh across [0, 1], the integral I can be written as

I =1

M

Mi=1

f[x(hi)]

g[x(hi)](2.7)

It is clear that the sampling now occurs according to the probability distribution.

This is called importance sampling. In Statistical Thermodynamics, one mostly

calculates properties which are averaged over many configurations corresponding

to the respective equilibrium ensembles. Unlike the example mentioned above, the

probability distribution function is known only within a proportionality constant.

The procedure for generating the configurations is given by the Metropolis algo-

rithm.

2.1.2 Metropolis Monte Carlo

In general the average of any quantity q(x) according to a probability distribu-

tion p(x) is calculated as

q(x)Avg = q(x)p(x)dx

p(x)dx (2.8)In MC simulations involving Statistical Thermodynamics, the properties that

are to be calculated are generally averaged according to the partition function (Hill,

1987). For example, the average of a quantity A depending upon the positions ofN

particles interacting via potential U is given as

A = A(rN)exp( [U(r

N)]kBT

)drNexp( [U(rN)]kBT )drN (2.9)


47/147


The quantity

A

is known as the ensemble average of the quantity A(rN). In

a brute force MC scheme, A(rN) would be calculated through giving all of the

N particles random positions, thereby generating a configuration and calculating

U(rN) for each of those configurations and hence calculating exp([U(rN)]kBT

). This

procedure would be repeated over many times and a statistical average would be cal-

culated. As far as importance sampling is concerned, the probability density is not

known exactly but only within a proportionality constant i.e. f(x) exp( [U(rN)]

kBT).

In such cases, the Metropolis algorithm (Frenkel and Smit, 2002; Allen and Tildes-

ley, 1989) is used. It involves generating particle configurations as a series of

Markov states i.e. a set of states in which a configuration is dependent only upon

the configuration that precedes it.

In this scheme, the sampling itself is done according to the probability distri-

bution rather than assigning the random positions their respective weights after the

sampling. Taking the canonical (NVT) ensemble as an example, the MC algorithm

can be stated as follows.

1. Starting from a configuration ofN particles, one particle is given a random

displacement ; (ri)new=(ri)old+r.

2. The change in energy associated with the random displacement given to the

ith particle U = UfinalUinitial is calculated. This is generally done in a way thatUfinal and Uinitial are measured with respect to the ith particle and not as cumulative

pair potential energies of the whole system. This makes the process computation-

ally less intensive.

3. IfU < 0 then the random move is accepted. This is quite straightforward

as the system goes to a favorable low energy state.


48/147

Chapter 2 31

4. IfU > 0 , the move is accepetd according to the probability exp( [U(rN)]

kBT).

This is done by generating a random number rand such that 0 < rand < 1 and

comparing its value with exp( [U(rN)]

kBT). Now, ifrand < exp( [U(r

N)]kBT

), the move

is accepted otherwise the starting configuration is restored.

5. The relevant property A(rN) is then sampled.

6. Steps are repeated enough number of times to get an accurate ensemble aver-

age.

It is important to reflect upon how the Metropolis MC scheme is different from

the one used to calculate the area of a circle. First things first, the calculation of the

area of a circle is a brute force MC scheme and does not involve any variance reduc-tion technique such as importance sampling. Secondly, the MC sampling scheme

for canonical ensemble is different from conventional importance sampling schemes

as the probability distribution is not known exactly but within a proportionality con-

stant, namely the Boltzmann factor exp( [U(rN)]

kBT). Therefore Metropolis algorithm

is used to generate a series of Markov states. In the present work only Metropolis

MC schemes are used. The general design is always on the lines of what is stated

above but advanced schemes are to be used wherever the standard routines are in-

sufficient to complete the task at hand. For, example the calculation of free energies

for both solid and fluid phases requires some additional procedures that need to be

incorporated in the standard MC schemes.


49/147


2.2 The isothermal-isobaric (NPT) ensemble

As the name suggests, this ensemble pertains to imposing the conditions of con-

stant pressure and temperature over a system consisting of a constant number of

particles. The statistical mechanics of the NPT ensemble is an extension of the

partition function of the NVT ensemble with an additional imposed condition.

2.2.1 Derivation of the NPT Partition function

Consider a system ofN identical particles at a constant volume and temperature.

The partition function is given by

Q(N , V , T ) =

1

3NN! L

0 ......L0 exp[U(r

N

)]drN

(2.10)

The assumption that the system is contained in a cubic box of side L and volume V

can be utilized to scale the coordinates by the box length.

si =riL

(2.11)

The partition function with scaled coordinates would be

Q(N , V , T ) =VN

3NN!

10

......

10

exp[U(sN)]dsN (2.12)

If this system is separated by a piston from an ideal gas reservoir with the sum

of the volumes of both systems being V0 and the total number of particles being

M, then the volume accessible to the M N ideal gas particles would be V0 V(Fig. 2.2). In this case, the partition function would be the product of the respective


50/147

Chapter 2 33

Figure 2.2: A Schematic showing a system of N interacting particles in a volume V. Thissystem can exchange volume with the ideal gas reservoir through the movable piston.


51/147


subsystem partition functions.

Q(N , M , V , V 0, T) =VN(V0 V)MN3MN!(M N)!

10

......

10


The assumption here is that the thermal de Broglie wavelength of the ideal gas

particles is the same as that of the N particles. The next step is to make the piston

free and hence allow the volumes of both the subsystems to fluctuate. In such a

scenario, the volume that leads to a minimum free energy will be the most probable.

In the limit V , M i.e. when the size of the ideal gas reservoir tendsto infinity, it can be easily assumed that small changes in the volume of the N-

particle system would not affect the pressure of large system. This leads to the

conclusion that large system acts as a manostat for the small N-particle system.

Now if (V0 V)(MN) is written as V(MN)0 [1 ( VV0 )](MN), then in the limit ofthe large system acting as a manostat i.e. V

V0 0, the LHopitals rule for limits of

the from 1 gives.

(V0 V)(MN) = V(MN)0 exp[(M N)V

V0] (2.14)

Since the number N is negligible compared to M, therefore

(V0 V)(MN) = V(MN)0 exp[V] (2.15)

where is the density of the ideal gas reservoir. For ideal gases, the equation of

state is P = and this helps in writing the partition function (2.13) as

Q(N, P, T) =P

3NN!

V0

VN exp[P V]dV10

......

10

exp[U(sN)]dsN

(2.16)

the P term outside the integrand is necessary to make the partition function,


52/147

Chapter 2 35

a probability distribution, dimensionless. The Gibbs free energy of the system is

given by

G(N, P, T) = kBT ln Q(N, P, T) (2.17)

2.2.2 Monte Carlo moves in the NPT ensemble

With the partition function of the NPT ensemble, it is obvious that the MC

scheme needs to be altered from that of the NVT system due to the introduction of

the additional variable i.e. volume. The probability that the N-particle system has

a volume V is given by

PN,P,T(V) =VN exp[P V] exp[U(sN)]dsNV0

0VN exp[P V]dV exp[U(sN)]dsN (2.18)

where V is the volume variable and s are the corresponding coordinates. The

probability density to find a particular configuration ofN particles at a given volume

V is approximated as

N(V, sN) VN exp[P V] exp[U(sN)] (2.19)

N(V, sN) = exp[[U(sN) + P V N

ln V]] (2.20)

For the NPT Monte Carlo Volume move, the volume of a system is changed from

say V to V through an increment of V. The value of V is chosen randomly

from the interval [Vmax, +Vmax] where V is the maximum allowed changein volume. Therefore, the volume move is accepted through the Metropolis method

according to the probability

exp[[(U(sN, V) U(sN, V)) + P(V V) N

lnV

V]] (2.21)

Volume moves are expensive and each volume move is done after a particular num-

ber of particle (translational) moves. It is important though that they are not per-


53/147


Figure 2.3: A schematic of the identity switch move. The total number of particles, volume

and the particle coordinates remain the same for both configurations.

formed periodically in order to maintain the symmetry of the Markov chain. Instead,

at every step there should be a probability that decides which of the two moves needs

to be performed.

2.3 The Semigrand ensemble

The Semigrand ensemble is one of the ways to study phase equilibria in mixtures.

The basis of this ensemble is that if the chemical potential of one of the compo-

nents in a mixture is fixed, then chemical potential of all other components can be

imposed through identity switches (Fig. 2.3) among the constitutive species (Kofke


54/147

Chapter 2 37

and Glandt, 1988).

2.3.1 The Grand-Canonical (,V,T) Ensemble

In the present work, the Grand-Canonical ensemble has not been used anywhere.

However, an understanding of the grand canonical ensemble helps us to understand

the semigrand ensemble better. The Grand canonical ensemble constrains the chem-

ical potential, volume and temperature of the system. With the chemical potential

fixed, the number of particles is allowed to fluctuate. The metropolis method can-

not compute quantities that depend explicitly depend on the configuration integralexp[U(rN)]drN e.g. Gibbs free energy. However, it can compute the difference

in free energies of two configurations. Grand Canonical Monte Carlo simulations

are based on this fact (Norman and Filinov, 1969; Adams, 1974).

To understand the statistical mechanics the system used for the derivation of the

NPT partition function will suffice. Eq.(2.13) gives the two system partition func-

tion as

Q(N , M , V , V 0, T) =VN(V0 V)MN3MN!(M

N)!

10

......

10


The only change here is that instead of exchanging volumes, the two systems

can exchange particles(Fig. 2.4). Moreover, when the particles are in the subvolume

V0 V, they act as ideal gas particles and otherwise i.e. in the subvolume V, theyinteract with each other. This implies that if we move the ith particle with reduced

coordinate si, from subvolume V0 V to the subvolume V, the potential energyfunctional changes from U(sN) to U(sN+1). Considering that, the partition function

including all possible configurations of the M particles within both the subvolumes


55/147


Figure 2.4: A Schematic showing thea system ofN interacting particles in a volume V. This

system can exchange particles with the ideal gas reservoir of volume V0 V. The volumeremains constant. A particle when inside the subvolume V, interacts with the interparticlepotential. Otherwise, it acts as and ideal gas particle. The volumes do not fluctuate.


56/147

Chapter 2 39

would be.

Q(M , V , V 0, T) =MN=0

VN(V0 V)MN3MN!(M N)!

10

......

10


On similar lines to the NPT derivation, considering the limit that the ideal gas

reservoir is much larger than the interacting system we have M , (V0 V) and therefore, M

(V0V) , where is density. Also, the density of an ideal gas

is related to its chemical potential as

= kBT ln(3) (2.24)

Therefore in the limit MN

, the partition function is given as

Q( , V , T) =N=0

exp[N]VN

3N!


And the probability density is

exp[N]VN

3N!exp[U(sN)] (2.26)

At this point the derivation of the semigrand partition function can begin.

2.3.2 The Semigrand partition function

The grand-canonical partiton function for an n-component mixture can be writ-

ten as

(1,....,n, V , T ) =

N1,N2,...,Nn

ni=1

qNii exp[iNi]

Ni!VN

exp[U(sN)]dsN

(2.27)

where N = i Ni and qi is the kinetic contribution to the partition function dueto the ith species. Also, the potential energy function inside the integrand is for the


57/147


total number of particles in the n-component system.

Imposing a constraint that the total number of partices N =

i Ni is fixed can

lead to the elimination of one of the Nis from the summation within the partition

function and that species can be written out. If that species is N1, the new partition

function is

(1,....,n, V , T ) =

N2,...,Nn

qN1 exp[1N]ni=1

[qiq1

]Niexp[(i 1)Ni]

Ni!VN

exp[U(sN)]dsN(2.28)

The next step is to generate a new partition function by multiplying both sides of

Eq.(2.28) by exp[1N]. We have = exp[1N]

(1,....,n, V , T ) =

N2,...,NnqN1

n

i=1[

qiq1

]Niexp[(i 1)Ni]

Ni!VN

exp[U(sN)]dsN

(2.29)

Consider the argument that different species in a mixture are all manifestations

of a certain basic type of particle and a species is defined as the kind of label that is

carried by each basic particle. These labels could be thought of as charges or sizes

or whatever property that can differentiate among different species. The semigrand

ensemble is pivoted around the flexibility to interchange these labels. In context of

Eq.(2.29) this argument can induce a variation in the distribution of particles among

the n indices of the summation sign. Each type of species Ni that earlier corre-

sponded to one term in the sum is now open to a change in the number of particles

that fall under it. Each of the total N particles can be of any one of the n types but


58/147

Chapter 2 41

Figure 2.5: A pictorial representation of the identity exchange principle through a simple

argument of permutations for a set of four particles. There are two particles each of iden-

tities red and yellow. It is clear that the total number of ways to arr

phase behaviour of oppositely charged nanoparticles: a study of binary nanoparticle superlattices

Documents