phase behaviour of oppositely charged nanoparticles: a study of binary nanoparticle superlattices
TRANSCRIPT
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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
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i
. To my grandparents
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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.
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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
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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
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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
4.5 Phase Diagram for Charge ratio 0.6; QBQA
= 0.6 . . . . . . . . . 1034.6 Phase Diagram for Charge ratio 0.8; QB
QA= 0.8 . . . . . . . . . 106
4.7 Phase Diagram for Charge ratio 1.0; QBQA
= 1.0 . . . . . . . . . 111
5 Conclusion 117
A Widoms test Particle insertion method 119
B Thermodynamic integration across chemical potential differences (s)121
References 125
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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
4.2 Coexistence points for the binary nanoparticle suspension with QBQA
=
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
4.3 Coexistence points for the binary nanoparticle suspension with QBQA
=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
4.4 Coexistence points for the binary nanoparticle suspension with QBQA
=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
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viii List of Tables
4.5 Coexistence points for the binary nanoparticle suspension with QBQA
=
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
4.7 Coexistence points for a binary nanoparticle suspension with QBQA
=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
4.8 Coexistence points for a binary nanoparticle suspension with QBQA
=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
4.9 Coexistence points for the binary nanoparticle suspension with QBQA
=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
4.11 Coexistence points for a binary nanoparticle suspension with QBQA
=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
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List of Tables ix
4.12 Coexistence points for a binary nanoparticle suspension with QBQA
=
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
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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
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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
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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
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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
4.2 Phase Diagram for a Binary Nanoparticle Suspension with QBQA
=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
4.3 Phase Diagram for a Binary Nanoparticle Suspension with QBQA
=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
4.4 Phase Diagram for a Binary Nanoparticle Suspension with QBQA
=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
4.5 Phase Diagram for a Binary Nanoparticle Suspension with QBQA
=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
4.6 Phase Diagram for a Binary Nanoparticle Suspension with QBQA
=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
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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
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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.
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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).
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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.
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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.
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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)
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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.
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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),
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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.
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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
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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
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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
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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 ).
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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.
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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.
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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
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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
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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).
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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.
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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
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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).
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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.
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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.
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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
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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
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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. .
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28 Monte Carlo Molecular Simulations
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
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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)
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30 Monte Carlo Molecular Simulations
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.
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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.
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32 Monte Carlo Molecular Simulations
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
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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.
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34 Monte Carlo Molecular Simulations
subsystem partition functions.
Q(N , M , V , V 0, T) =VN(V0 V)MN3MN!(M N)!
10
......
10
exp[U(sN)]dsN (2.13)
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,
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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-
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36 Monte Carlo Molecular Simulations
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
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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
exp[U(sN)]dsN (2.22)
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
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38 Monte Carlo Molecular Simulations
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.
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Chapter 2 39
would be.
Q(M , V , V 0, T) =MN=0
VN(V0 V)MN3MN!(M N)!
10
......
10
exp[U(sN)]dsN (2.23)
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!
exp[U(sN)]dsN (2.25)
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
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40 Monte Carlo Molecular Simulations
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
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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