NOVEL PHASES IN POLYHEDRAL NANOPARTICLES: MIXTURES AND SPATIAL
CONFINEMENT
A Dissertation Presented to the Faculty of the Graduate School
of Cornell University in Partial Fulfillment of the Requirements for the Degree of
Doctor of Philosophy
by Mihir Ramchandra Khadilkar
August 2015

c 2015 Mihir Ramchandra Khadilkar ALL RIGHTS RESERVED

NOVEL PHASES IN POLYHEDRAL NANOPARTICLES: MIXTURES AND SPATIAL CONFINEMENT
Mihir Ramchandra Khadilkar, Ph.D. Cornell University 2015
Colloids present an interesting experimental system to study fundamental scientific problems as well as to tackle technological challenges, through novel material design. Several control parameters like size, shape, inter-particle interactions, assembly-geometry (including dimensionality) and external fields (among many others) can result in a great variety in morphologies and material properties. In particular, polyhedral nanoparticles are potentially powerful candidates with a rich phase behavior and availability of robust experimental methods for their synthesis. Our aim in this thesis is to understand, through computer simulations, various aspects of polyhedra phase behavior.
In particular, we study a specific case of binary mixtures of polyhedra called binary tessellating mixtures in Chapter 2. The motivation here is to study if these superstructures are generated from the geometrical condition of spacefilling i.e. tessellation, without the use of any enthalpic interactions (which tend to be harder to control in experiments). As we see in Chapter 2, pure entropic self-assembly of these mixtures fails to reach tessellated phase due to kinetic barriers, which can be alleviated by small targeted enthalpic interactions.
We further explore the wider problem of self-assembly of binary polyhedral mixtures in Chapter 3 to understand the generic predictive rules that can help guide experimental efforts. We find that the mixture miscibility (a critical criterion for novel superstructures) is strongly determined, among other factors, by

the difference between order-disorder transition pressure (∆ODP) of the individual polyhedra in the mixture. We also propose a general qualitative roadmap for the mixture phase behavior.
In chapter 4, using the guiding rules discovered while studying mixtures, in combination with novel plastic crystalline ‘mesophases’ exhibited by a subset of polyhedra, we develop a design scheme that allows for the formation of ordered mixtures without introducing any enthalpic interactions. These so-called Mixed Rotator mesophases (MRMs) form purely from an entropic self-assembly and are stable for a large range of volume fractions.
Apart from shape bi-dispersity (mixtures), we also investigate the effect of geometrical confinement on polyhedral self-assembly in Chapter 5 and show that a parallel-plate confinement leads to many novel phases that are not seen in bulk, through the case of four representative polyhedra from the truncated cube family. We conclude with a summary of our findings and a discussion of currently prevalent research directions.

BIOGRAPHICAL SKETCH Mihir Khadilkar was born in Pune, India in 1986. Growing up in Pune, he graduated high school in 2004. He completed his undergraduate education at Indian Institute of Technology Bombay at Mumbai in 2009, with a Bachelor of Technology in Engineering Physics. During his time as an undergraduate, he was exposed to the world of research through two summer research opportunities in Ireland and France.
He moved to Ithaca, NY in fall of 2009 in the Department of Physics at Cornell University with Cornell Graduate Fellowship. After a short research experience in other projects, he started working on problems in soft condensed matter, in Prof. Fernando Escobedo’s research group in the Department of Chemical and Biomolecular engineering.
iii

To Aai, Baba and Ishita
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ACKNOWLEDGEMENTS
It is often said that pursuing a Ph.D. is more like a marathon rather than a short sprint. There are long periods of lull, exhaustion, disappointment, perseverance and hopefully success. But all of this is extremely difficult without the support of those around you. I want to take this moment to acknowledge their help, support and guidance.
First and foremost, I would like to thank my advisor Prof. Fernando Escobedo, who has been extremely helpful during this entire process. His keen intuition in our research problems has guided my efforts well. He gave me complete freedom to work at my own pace, helped me when I was stuck and was constantly available in case I needed his guidance. His style of research and scientific inquiry in general has greatly influenced my own and I am deeply indebted to him for that.
I would also like to thank my committee members Prof. Veit Elser and Prof. Itai Cohen, whom I have often consulted regarding my projects and sought their advice. I also acknowledge U.S. National Science Foundation (Grant No. CBET 1033349 and CBET 1402117) and U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering (Grant No. ER46517) for their funding.
My colleagues in the Escobedo group have also been greatly helpful. I would like to especially thank Umang Agarwal and Carlos Avendan˜ o, who have been excellent mentors during my early stages in the group, getting me started in the field. I have cherished interacting with all the group members including Poornima, Vikram, Pooja, Sushmit, Salomon, Stacey and Beth, on both academic and non-academic topics.
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My friends from the Physics department have all been really helpful, especially in my early years in graduate school. I would like to thank Shivam, Hitesh, Kartik, Inˆ es, Jesse, Thomas and Kyung Min for their friendship and support during trying times of graduate school. My social life at Cornell was shaped greatly by my friends here and my housemates Anirikh, Shantanu, Umang, Praveen and Rajesh. My time at Cornell was greatly fulfilling and enjoyable in spite of all the hurdles because of them.
Finally, I am truly indebted to my family, who have provided continual emotional support. My brother, sister-in-law, my in-laws have all constantly encouraged me in this journey. My parents been a constant source of strength, motivation and inspiration for me throughout the years and I cannot thank them enough. Lastly, I am tremendously thankful to have Ishita by my side, during this journey and for the decades to come.
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TABLE OF CONTENTS

Biographical Sketch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

iii iv v vii x xi

1 Introduction 1.1 Length scales and interaction in colloids . . . . . . . . . . . . . . . 1.2 Synthesis, theory and simulations . . . . . . . . . . . . . . . . . . . 1.3 Anisotropy and polyhedral nanoparticles . . . . . . . . . . . . . . 1.4 Interactions: Entropic vs. Enthalpic Self-assembly . . . . . . . . . 1.5 Mixture phase behavior: Mixing Vs. Packing entropy . . . . . . . 1.6 Organization of the thesis . . . . . . . . . . . . . . . . . . . . . . .

1 2 3 4 6 8 9

2 Binary Tessellating Compounds

11

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3.1 Perfect hard polyhedrons . . . . . . . . . . . . . . . . . . . 17

2.3.2 Polybead systems . . . . . . . . . . . . . . . . . . . . . . . . 21

2.4 Results for perfect polyhedrons . . . . . . . . . . . . . . . . . . . . 25

2.4.1 Mixture 1: Rectified Cubic honeycomb (RCH) . . . . . . . 25

2.4.2 Mixture 2: Alternated Cubic Honeycomb (ACH) . . . . . . 27

2.4.3 Mixture 3: Truncated Cubic Honeycomb (TCH) . . . . . . 29

2.4.4 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.5 Results with Polybead Models . . . . . . . . . . . . . . . . . . . . . 35

2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3 Generic Binary Mixtures

42

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.2 Polyol Process and choice of systems studied . . . . . . . . . . . . 44

3.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.4.1 Cubes + Spheres (CS) Mixture . . . . . . . . . . . . . . . . 53

3.4.2 Cubes + Truncated Octahedra (CTO) Mixture . . . . . . . . 56

3.4.3 Cuboctahedra + Cubes (COC) Mixture . . . . . . . . . . . 58

3.4.4 Cuboctahedra + Truncated Octahedra (COTO) mixture . . 59

3.5 Discussion of general trends and comparison to other systems . . 63

3.6 Conclusions and a roadmap for equimolar phase behavior . . . . 67

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4 ODP-tuning and Mixed Rotator Mesophases 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Role of relative size-ratios and mesophases . . . . . . . . . . . . . 4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Importance of ODP-tuning . . . . . . . . . . . . . . . . . . . . . . . 4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

72 72 73 75 79 83

5 Phase behavior of polyhedral nanoparticles in parallel plate confine-

ment

85

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.2 Shapes, Model and methodology . . . . . . . . . . . . . . . . . . . 87

5.2.1 Order parameters . . . . . . . . . . . . . . . . . . . . . . . . 88

5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.3.1 Cubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.3.2 TC4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.3.3 Cuboctahedra (COs) . . . . . . . . . . . . . . . . . . . . . . 98

5.3.4 Truncated Octahedra (TOs) . . . . . . . . . . . . . . . . . . 101

5.3.5 Remarks on global trends . . . . . . . . . . . . . . . . . . . 105

5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6 Conclusions and relevance to experiments

109

6.1 Summary of key results . . . . . . . . . . . . . . . . . . . . . . . . 109

6.2 Relevance to experiments: study of kinetics . . . . . . . . . . . . . 111

6.3 Currently prevalent ideas and open questions . . . . . . . . . . . 114

A Supplementary information on MRM phase behavior

119

A.1 Shape definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

A.2 Isotropic branch of Equation of State (EoS) for different shapes . . 120

A.3 Simulation procedure for binary mixtures with CO, TO and TC4 . 121

A.4 Equations of state for main binary mixtures studied . . . . . . . . 122

A.5 Ternary Mixture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

A.6 Simulation details for mixtures of spheres and cubes . . . . . . . . 124

A.7 Metrics for the analysis of configurations . . . . . . . . . . . . . . 126

A.7.1 P4, Q4, Q6 order parameters . . . . . . . . . . . . . . . . . . 126

A.7.2 MRM structure determination . . . . . . . . . . . . . . . . 126

A.7.3 Orientational scatterplots . . . . . . . . . . . . . . . . . . . 130

B Supplementary information on polyhedra in parallel plate confine-

ment

133

B.1 Cubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

B.2 TC4s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

B.3 COs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

B.4 TOs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

viii

Bibliography

139

ix

LIST OF TABLES 2.1 Order-disorder transition pressures (ODPs) for polyhedra in tes-
sellating compounds . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.1 Reference data for polyhedra in generic binary mixtures studied 49 4.1 Summary of miscibility range data for different size ratios in
COTO mixture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 A.1 Reference data for shapes studied for MRM phase behavior . . . 119
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LIST OF FIGURES

1.1 Cartoon depicting pure entropic self-assembly . . . . . . . . . . . 8

2.1 Generic phase diagrams for binary mixtures . . . . . . . . . . . . 2.2 Binary tessellating compounds studied . . . . . . . . . . . . . . . 2.3 polybead models for tessellating compounds . . . . . . . . . . . . 2.4 Results for Rectified Cubic Honeycomb (RCH) . . . . . . . . . . . 2.5 Results for Alternated Cubic Honeycomb (ACH) . . . . . . . . . 2.6 Results for Truncated Cubic Honeycomb (TCH) . . . . . . . . . . 2.7 Cluster analysis for tessellating compounds . . . . . . . . . . . . 2.8 Snapshots for clusters in tessellating compounds . . . . . . . . . 2.9 Results for polybead model for RCH mixture . . . . . . . . . . . . 2.10 Results for polybead model for ACH mixture . . . . . . . . . . . 2.11 Sketch of free energy landscape for mixture self-assembly . . . .

14 16 22 26 28 30 33 33 36 37 39

3.1 Polyhdera from polyol process . . . . . . . . . . . . . . . . . . . . 3.2 Generic binary polyhedral mixtures studied . . . . . . . . . . . . 3.3 Orientational scatterplots for Cuboctahedra-Truncated Octahe-
dra mixture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Results for Cubes-Spheres mixture . . . . . . . . . . . . . . . . . . 3.5 Results for Cubes-Truncated Octahedra mixture . . . . . . . . . . 3.6 Results for Cubes-Cuboctahedra mixture . . . . . . . . . . . . . . 3.7 Results for Cuboctahedra -Truncated octahedra mixture . . . . . 3.8 Qualitative phase diagrams for binary polyhedral mixtures . . . 3.9 Overlap of phase diagrams for CS, CTO and COC mixtures . . . 3.10 Tentative road map for binary mixture phase behavior . . . . . .

44 47
52 54 57 60 62 64 65 69

4.1 Phase diagram, scatterplots and snapshots for three main mixtures exhibiting MRM behavior . . . . . . . . . . . . . . . . . . . . 76
4.2 Plots showing the effect of changing the mesophase composition in COTO mixture . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.3 Phase diagram for Cubes-Spheres mixture at different size ratios 81

5.1 Shapes studied in confined geometry . . . . . . . . . . . . . . . . 87 5.2 Phase diagram for cubes in confinement . . . . . . . . . . . . . . 93 5.3 Snapshots for cubes in confinement . . . . . . . . . . . . . . . . . 94 5.4 Phase diagram for TC4s in confinement . . . . . . . . . . . . . . . 96 5.5 Snapshots for TC4s in confinement . . . . . . . . . . . . . . . . . 97 5.6 Phase diagram for COs in confinement . . . . . . . . . . . . . . . 98 5.7 Snapshots for COs in confinement . . . . . . . . . . . . . . . . . . 100 5.8 Phase diagram for TOs in confinement . . . . . . . . . . . . . . . 102 5.9 Snapshots for TOs in confinement . . . . . . . . . . . . . . . . . . 104

A.1 EoS comparison for different shapes at ODP-matched ratio . . . 121

xi

A.2 Equation of state for the equimolar COTO mixture . . . . . . . . 122 A.3 Equation of state for the equimolar TC4TO mixture . . . . . . . . 122 A.4 Equation of state for the equimolar TC4CO mixture . . . . . . . . 123 A.5 Equation of state for the ternary mixture . . . . . . . . . . . . . . 124 A.6 Snapshot for MRM phase in ternary mixture . . . . . . . . . . . . 124 A.7 Order parameter plots for COTO mixture . . . . . . . . . . . . . . 127 A.8 Order parameter plots for TC4TO mixture . . . . . . . . . . . . . 127 A.9 Order parameter plots for TC4CO mixture . . . . . . . . . . . . . 128 A.10 Plot showing distributions of different order parameters for ref-
erence phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 A.11 Plots showing fractions of crystal structure present in COTO
mixture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 A.12 Plots showing fractions of crystal structure present in TC4TO
mixture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 A.13 Plots showing fractions of crystal structure present in TC4CO
mixture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 A.14 Orientational scatterplots for the main three mixtures . . . . . . . 132 B.1 Order parameter plot for cubes at H*=1.1 . . . . . . . . . . . . . . 133 B.2 Order parameter plot for cubes at H*=1.3 . . . . . . . . . . . . . . 134 B.3 Order parameter plot for TC4s at H*=2.0 . . . . . . . . . . . . . . 135 B.4 Order parameter plot for COs at H*= 1.6 . . . . . . . . . . . . . . 136 B.5 Order parameter plot for COs at H*= 2.8 . . . . . . . . . . . . . . 136 B.6 Order parameter plot for TOs at H*= 1.9 . . . . . . . . . . . . . . 137 B.7 Order parameter plot for TOs at H*= 2.2 . . . . . . . . . . . . . . 138
xii

CHAPTER 1 INTRODUCTION
The work in this thesis is focused on colloidal particles, which are particles within a certain size range (typically from 1 nm to 1 micron) dispersed in a solvent. Such particles fall in the interesting regime of ‘mesoscopic’ systems, wherein the particle size is intermediate between microscopic systems like atoms and macroscopic systems like books or bricks. Colloids are ubiquitous in common everyday objects including milk, butter, jelly, toothpaste, colored glass among many others. Over the last four decades or so, colloids have emerged as an important class of materials both in terms of applications to everyday products, as well as being excellent model systems to study the atomic world. Ability to manipulate material chemistry one atom at a time would be highly desirable for the design of versatile functional nanomaterials, but is hardly tractable at the moment. Colloids on the other hand, have shown great promise as building blocks to controllably produce a great variety of structures. Further, the colloidal particles and their ordered structures have been shown to be extremely useful towards functional applications such as chemical sensors[1], catalysts[2, 3], optics[4] and solar cells[5]. Due to their versatility, one can fine-tune their material properties such as crystal structure, electrical conductivity, the band gap in semiconductor nanoparticles, the paramagnetic behavior of magnetic nanoparticles[6, 7], and the bandgap in photonic crystals [8, 9].
Colloidal systems are interesting for a number of reasons. Microscopic systems like atoms and electrons often require quantum mechanics for their full description. Mesoscopic systems however, are two or three orders of magnitude bigger than these microscopic systems, and hence are easier to analyze. In
1

particular, a description without quantum mechanics using only classical statistical physics is enough to understand a majority of their physical phenomena. These problems are thus more tractable in theory, simulation as well as experiments. Experiments involving mesoscopic systems like colloids often allow the visualization and tracking of the structure and motion of the particles using optical microscopy, confocal microscopy[10, 11], and in certain cases Transmission Electron Microscope (TEM) and Scanning Electron Microscope (SEM). This facilitates comparisons between experiments and theory or simulation. Easier manipulation also opens doors to more complex self-assembly techniques like hierarchical self-assembly[12]. Hierarchical self-assembly has been known to be a common route adopted by nature to create many useful biological materials[13–15] and holds promise for colloids as well.
1.1 Length scales and interaction in colloids
Roughly, as described above, the length scales we deal with while studying colloids are 1 nm to 1 micron. This scale is important because of the relative strength of different forces at play. One of the most important features in colloids is Brownian motion, which is their constant random motion - first observed by Robert Brown in 19th century. This phenomenon is caused by an instantaneous imbalance of the forces exerted by the solvent molecules on the colloidal particles. Equally important is the fact that other forces like gravity and contact forces tend to be negligible compared to the Brownian forces, unlike in the case of granular materials[16]. Brownian motion is an important factor in the assembly of colloids into crystals since it is this random motion that allows them to explore the configurational phase space and access the thermodynamically
2

favorable structure.
Although most colloidal suspensions are solutions of (relatively) large colloidal particles in smaller solvent particles, in many cases they can be described without an explicit solvent with just an effective interaction. These interactions could be electric/magnetic interactions, hard-core repulsions, dispersion forces (due to dipole moments), depletion interaction[17, 18] (which is a purely entropic effect caused by depletants dispersed in the solvent), among others. The relative strength of each of the forces radically depends on the particular material, size, shape and the mode of preparation. Often, the colloidal particles need to be coated with stabilizing agents like charged ions or polymers. All these factors also decide the ways in which these colloidal systems could be modeled in simulations (which is the focus of this work).
1.2 Synthesis, theory and simulations
Methods to synthesize nanoparticles have been known since as early as 1960s[19]; however, efforts to gain a precise control over the structure, size, shape and inter-particle interactions still drive current research in many directions. Initially, the techniques were restricted to simplest spherical nanoparticles produced via sedimentation, emulsion polymerization[20] or similar methods[3, 21, 22]. Some of the earliest spherical colloids synthesized were silica[23, 24], polymethyl methacrylate (PMMA)[25] and polystyrene[26]. However, methods to synthesize non-spherical nanoparticles including cubes, rods and discs[27–31] were soon developed. Development of new physical methods fueled the field of synthesis even further, including selective crystalliza-
3

tion[32, 33], lithography-based methods[34–36] and a variety of other techniques[37–40].
The efforts to understand colloidal phase behavior started even before the development of controlled experimental synthesis methods, initially through theory and later through computer simulations. Some of the early theoretical studies dealt with depletion interaction[41], hard rods[42] and spheres[43]. With the advent of computer simulations, theoretical results for these and many other systems could be confirmed and a much larger class of systems were now amenable to analysis[44–50].
1.3 Anisotropy and polyhedral nanoparticles
As described earlier, the initial focus on spherical nanoparticles soon gave way to anisotropic nanoparticles since they entailed more complex interactions and allowed greater diversity in structure and phase behavior[51]. With the ability to synthesize various anisotropic colloids, including cubes[52, 53], rods[54–56], dumbbells[57,58], janus particles [59,60], tripod and tetrapods [61], striped particles[62] and many others, the phase space of available building blocks increased rapidly. Along with the advent of all these building blocks, came the efforts to predict the different thermodynamic phases that these particles can exhibit. Apart from the common crystalline structures like the face-centered and body-centered cubic (FCC and BCC) shown by simpler building blocks, many of the anisotropic shapes discussed above also showed nematic and smectic liquid crystalline behavior.
More recently, experimental methods were developed that could synthe-
4

size a wide range of polygonal and polyhedral particles including cubes and truncated cubes[37–39, 63, 64], octahedra[54, 65–68], tetrahedra[69–71], icosahedra[72, 73] and triangular bipyramids[74–76]. These polyhedral shapes are also anisotropic like the shapes mentioned earlier , but they exhibit higher rotational symmetry[77].
From the perspective of theory and simulation, some of the early research was motivated by longstanding problems in the field of discrete geometry. Famous puzzles like Kepler’s conjecture[78] (which dealt with the densest packing of spheres) and Ulam’s conjecture[79] were not just of fundamental nature, but they also had a practical impact in physics and material science. This is because the densest packing of an arbitrary anisotropic object generally tends to be a strong contender for its thermodynamically stable structure at high densities. Hence the knowledge of densest packings can guide computer simulations of phase behavior. In this context, many polyhedra were initially studied for their densest packings[80–85] rather than self-assembly. However, simulation studies of self-assembly behavior at different packing densities are more relevant to the soft-matter experiments and have revealed a very rich variety of phases [68, 77, 86–94], including crystals, disordered solids, plastic crystals and liquid crystals. Of particular importance is the fact that the combination of shape anisotropy and rotational symmetry allows many of the polyhedra to exhibit an intermediate ‘mesophase’[77] , with either partial translational or orientational order. This has a profound impact on building bigger, more complex structures from these polyhedral building blocks, as we will see in the upcoming chapters.
5

1.4 Interactions: Entropic vs. Enthalpic Self-assembly
While performing computer simulations of colloids, the model details for the colloidal particle are crucial in connecting with a particular physical scenario. The inter-particle interactions modeled in the description vitally affect the selfassembly and phase behavior of the particles.
Many factors influence how colloidal particles interact with each other. The solvent in which they are dispersed acts as a ‘mediator’ between their interactions and can often interact with the colloid in complicated ways. Thankfully, many a times, one can eliminate solvent degrees of freedom by representing them in terms of an ‘effective’ potential. The size, shape, electrical charge of the particles as well as the presence of external fields, all decide the full nature and extent of interactions between colloidal particles. Often, one can introduce socalled ‘depletion’ forces[17,18] by adding a small amount of free, non-adsorbing polymer to a colloidal suspension. These forces are purely entropic in nature and are caused by exclusion effects between polymer and the colloidal particles. The total effective interaction in the model, is thus determined by the relative strength of all different factors mentioned above.
Depending on the experimental conditions, size, shape and chemical composition, different effective potentials can be used to model a colloidal particle. For example, if the interaction tends to be extremely short-ranged and sharply repulsive, the colloidal particles essentially behave as non-interacting rigid ‘solids’. This is termed as hard-core potential. While it may seem an oversimplification, hard-core potential in many cases provides an accurate description. In case of polyhedral nanoparticles, the effects of shape anisotropy are
6

accurately represented by the non-overlap condition between any of the facets of the two polyhedral particles. This type of interaction is then purely entropic since enthalpy plays no role. There are of course scenarios where this description is not enough and one needs to take into account the enthalpic forces to accurately describe preferential attraction or repulsion as in Janus particles, or electrostatic forces in charged colloids.
In case of polyhedral nanoparticles, many of the experimental synthesis methods make use of capping agents, coatings or polymer ‘fur’ that makes them behave like particles with nearly hard-core interaction[95, 96]. Hence, in our analysis to follow, we extensively use the hard-core interaction model since this description is quite accurate for many experimental systems of interest. Further, by studying polyhedra as hard anisotropic particles, we isolate the effects of shape anisotropy from those of enthalpic interactions that may be present in more complicated models. This is enlightening since by understanding the effects of entropy, one can systematically add enthalpic effects later. Such a self-assembly driven by hard-core interactions only is termed as purely entropic self-assembly since enthalpic interactions are absent. The equilibrium phases at a given state of thermodynamic variables favor states with large entropy (number of possible configurations). For example, at low volume fractions and low pressure when there is an abundance of free volume, the entropy is maximized in a disordered phase (see Figure 1.1), since particles can then access all possible orientations. At high volume fractions and high pressure however, when free volume is scarce, the system can access more configurations by adopting an ordered phase.
7

Figure 1.1: Cartoon depicting pure entropic self-assembly
1.5 Mixture phase behavior: Mixing Vs. Packing entropy
One of the main themes of this work is exploring the variety and complexity in superstructures made up of colloidal building blocks. One of the simplest ways towards creating that variety is to use multiple components[97–101], since a competition between different component phases often leads to even richer phase behavior. However, such assemblies often require optimal tuning of the inter-particle interactions to ensure the stability of the binary superstructure.
In a purely entropic self-assembly, a mixture of two shapes often leads to phase separation at high enough volume fractions. Commonly, most phase transitions are affected by both entropic and enthalpic effects, and it is the competition between these effects that usually drives the transition upon changes in a thermodynamic variable (say, pressure or temperature). However, in a purely entropic self-assembly for a mixture, two kinds of entropic contributions compete with each other and hence drive the transition. Qualitatively, one can split the entropic contributions in a mixture into two components: mixing entropy,
8

which denotes the additional configurations accesible to the system due to mixing of two species and packing entropy, which denotes the additional possible configurations due to ordering of the available particles. This competition between the two kinds of entropy forms the central topic of discussion throughout this thesis.
1.6 Organization of the thesis
This thesis is organized in the following manner. Starting our inquiry into polyhedral binary mixtures, we focus our attention on the so-called tessellating binary mixtures. These are binary mixtures of polyhedra that tessellate (fill space) in three dimensions. These are ideal candidates towards exploring formation of binary superstructures, without the need of tuning the enthalpic interactions. We explore this self-assembly in Chapter 2. After understanding these systems, we move on to exploring more general cases of binary polyhedral mixtures (the ones that do not tessellate space) in Chapter 3. The aim of this study is to develop some of the guiding rules that determine the mixture phase behavior given the phase behavior of the constituent polyhedra. Having understood these generic principles, we propose a strategy towards creating ordered binary superstructures from polyhedral nanoparticles within a pure entropic perspective in Chapter 4. Using the partially ordered ‘mesophases’ of constituent polyhedra and tuning their size ratio to tune individual order-disorder transition pressures, we demonstrate binary polyhedral superstructures that are stable over a large region in phase space. Since geometrical confinement is also an important factor determining phase behavior, we lastly study the effect of parallel-plate confinement on monodisperse polyhedra in Chapter 5. We con-
9

clude with a concise review of our findings, an assessment of their impact on experimental studies and a discussion of currently prevailing ideas and open questions in this field in Chapter 6.
All chapters are written with the objective of being independent treatises on specific problems. They are of course tied together in a central theme and progression of ideas. But to facilitate the reader interested in particular parts of the thesis, each chapter has introductory discussion leading up to the problem at hand.
10

CHAPTER 2 BINARY TESSELLATING COMPOUNDS
The1 self-assembly of polyhedral particles has been a topic of interest in both experimental and simulation studies due to its potential to help engineer novel materials from colloidal nanoparticles. An important extension to the study of single species of polyhedral particles is the case of binary mixtures. Mixtures that tessellate space are particularly interesting because they are expected to form high-pressure ordered structures. Here, we study 3 such binary tessellating mixtures; namely, cuboctahedra + octahedra (Mixture 1), octahedra + tetrahedra (Mixture 2), and truncated cubes + octahedra (Mixture 3). We use Monte Carlo methods to first determine their phase behavior when driven by hard-core interactions (i.e., entropic self-assembly). We observe that upon gradual compression of the isotropic system, none of the three cases exhibits a spontaneous ordering into the expected tessellated structure: Mixtures 1 and 2 form a glassy disordered state that is shown to be metastable with respect to the tessellated phase via interfacial simulations; Mixture 3 demixes into a disordered phase and an unusual ordered phase where truncated cubes arrange in a cubic lattice while the octahedra remain disordered occupying interstitial pockets. Using polybead models for Mixtures 1 and 2, we show that the large free-energy barrier that precludes the spontaneous nucleation of the tessellating structure from the isotropic state can be overcome by introducing favorable enthalpic interactions. Our results allow identifying some relations between properties of individual species and the phase behavior of their mixtures, providing a first step toward a ‘chemistry’ of polyhedral compounds, while also raising key questions
1Reproduced with permission from ”Mihir R. Khadilkar and Fernando A. Escobedo. Selfassembly of binary space-tessellating compounds. J. Chem. Phys., 137(19):194907, 2012”. Copyright [2012], AIP Publishing LLC.
11

regarding the kinetics of the pseudo ‘reactions’ involved.
2.1 Introduction
During recent years, there have been rapid advances of particle synthesis techniques[37, 69, 102], targeted towards various applications such as optics[4], chemical sensors[1] and solar cell technologies[5]. These techniques have allowed the synthesis of particles with a wide variety of sizes, shapes, patterns, compositions, symmetries, and functionalities. Many of these synthesis methods involve the self-assembly of particles that act as building blocks to desired structures, much like atoms into different molecules. Unlike atoms, there is a significant freedom and control over the properties of the individual colloidal building blocks; this vast table of pseudo ‘elements’ could be used to produce an effective ‘supra chemistry’ of colloids.
For the creation of novel colloidal building blocks, a wide variety of techniques have been previously suggested[51], based on changes of the shape, composition, patterning, patchiness, roughness, chemical ordering, and faceting of particles. Out of those, tuning just the particle shape and composition provides a simple yet rich playground given the variety of self-assembled structures that can thus be achieved [51, 80, 103].
In the past, the relation between shape anisotropy and phase behavior has been elucidated[77, 87] for single-component space-filling hard polyhedral particles. Recent experiments [68, 104, 105] also demonstrate that polyhedral particles are promising candidates for targeted self-assembly into superlattices. Building onto previous studies, in this work we consider the phase behavior
12

of three special cases of binary mixtures of polyhedral particles, which form so-called ‘Convex Uniform Honeycombs’; i.e., they are mixtures of two convex uniform polyhedra that together tessellate space. These systems are interesting for several reasons. Towards the goal of exploring the ‘chemistry’ of these building blocks (as previously mentioned), these systems represent a natural and simple extension of the single-component polyhedron systems. They can also lead to useful novel ordered structures, especially if one of the components is selectively etched off (to create an open matrix) or replaced by another functional material[106]. At high pressures, the configuration that packs hard-core particles most efficiently is entropically favorable, but the relevant structure is not always known or regular. Since the systems of interest are space-filling or tessellating, we know a priori what the close-packed configuration is. Thus, an obvious question to try to answer is whether this configuration is kinetically and/or thermodynamically favorable.
Consider for concreteness a mixture of hard-core polyhedral particles A and B. In non-tessellating binary mixtures, particles crystallizing into incompatible lattices will tend to phase-separate at high pressures where the packing entropy overrides the mixing entropy, leading to an A-rich crystal A(c) and a B-rich crystal B(c). Thus, the qualitative phase-behavior in these cases is to some extent predictable and a typical phase diagram may look as shown in Figure 2.1 a. Depending on the relative size and shape similarity between A and B, some amount of blending may occur wherein, e.g., A(c) may dissolve some amount of B. Also, if the melting or order-disorder transition pressures (ODPs) of A(c) and B(c) are noticeably different, isotropic-solid two-phase regions may occur at intermediate pressures and compositions, before the A(c) + B(c) solid-solid twophase region ensues at high pressures. In contrast, in tessellating mixtures the
13

(a) (b)
(c) (d)
Figure 2.1: Sketches of generic phase diagrams for a binary mixture (of particle types A and B) exhibiting limited solid- solid miscibility forming an A-rich crystal A(c) and B-rich crystal B(c) at high pressures for: (a) when no compound is formed, (b), (c), and (d) different scenarios when an ‘entropic’ compound AB(c) is formed by the space-tessellating nature of the mixture. The isotropic phase is denoted as ‘I’ and a solid solution as AB(ss). Shaded regions correspond to twophase states. The shape of the phase diagram depends on the relative values of the melting pressures for A(c), B(c), and AB(c). space-filling composition provides another strong candidate for a distinct highpressure phase, the AB(c) compound, which can change drastically the phase landscape as illustrated in Figure 2.1. Depending on the lowest stability pressure of AB(c) relative to ODPs of A(c) and B(c), different scenarios may occur upon decompressing the AB(c) compound. For example, AB(c) may directly melt into a homogenous liquid as in 2.1 b, or first decompose into a liquid-solid two-phase state as in 2.1 c. If the crystalline structure of one of the components, say B, is the
14

same in B(c) and in the AB(c) compound, then a solid solution AB(ss) may form at high pressures for intermediate compositions, and the AB(c) compound arise only at some composition and pressure range, as in 2.1 d. Effectively, Figures 2.1 b-d suggest that A(c) and B(c) figuratively react to form the AB(c) compound: A(c) + B(c) → AB(c)
In this work, we are not concerned with mapping the complete phase diagram of tessellating A+B mixtures (as in Figure 2.1 ) but only to determine the behavior of the stoichiometric composition of the AB(c) compound (indicated by the red arrows in Figure 2.1). In particular, we aim to get insights on the apparent kinetic barrier to self-assembly of the tessellated phase, the presence or lack thereof of the mesophases[77], and the effect of particle packing fraction and selective enthalpic interactions on phase behavior.
2.2 Systems
As described earlier, we chose 3 examples of binary mixtures of polyhedral particles that tessellate space. In the literature, the tessellating structures they form are called ‘convex uniform honeycombs’. Mixture 1 comprises cuboctahedra and octahedra in a 1:1 ratio and forms the ‘rectified cubic honeycomb’ (RCH). Mixture 2 comprises octahedra and tetrahedra in a ratio 1:2 and forms an ‘alternated cubic honeycomb’ (ACH). Mixture 3 comprises truncated cubes and octahedra in a ratio 1:1 and forms the ‘truncated cubic honeycomb’ (TCH). These mixtures are chosen on the basis of being made by polyhedra which can be readily synthesized via well-known approaches[68, 107, 108], and have the fewest number of facets (i.e., they could be considered the ‘simplest’ binary mixtures
15

that tessellate space). As shown in Figure 2.2 , in each of these mixtures the length of the edge of both polyhedra is equal to each other, while the ratios of volumes (large to small component) are 5, 4, and 28.85, respectively.
Figure 2.2: Convex Uniform Honeycombs: a) Rectified Cubic Honeycomb (RCH). Purple = Cuboctahedra, Green = Octahedra. b) Alternated Cubic Honeycomb (ACH). Green = Octahedra, Orange = Tetrahedra. c) Truncated Cubic Honeycomb (TCH). Green = Truncated Cubes, Red = Octahedra.
Besides the ACH, Mixture 2 can also tessellate space in a less symmetric arrangement called gyrated alternated cubic honeycomb (GACH) which has layers rotated 60 degrees so that half the edges have neighboring rather than alternating tetrahedra and octahedra[109]. Referring back to Figure 2.1, we interpret this case as an AB(c) compound that is degenerate, which should increase the entropy (and stability) of the ordered phase. For the ensuing simulations that start off with an assumed crystal structure, we will only consider the ACH for Mixture 2 in view that (i) ACH and GACH are very similar and expected to have nearly identical free energies and coexist together (in large systems), and (ii) one can very trivially introduce an enthalpic bias toward the ACH via electrostatic interactions (see Subsection 2.3.2). Of course, simulations that start from a disordered state do not make any assumption about the ordered phase
16

that can be formed.

2.3 Methodology

2.3.1 Perfect hard polyhedrons

Mapping of equation of state

To map out the equation of state of each of the mixtures, extensive expansion and compression Monte Carlo (MC) runs at constant pressure and particle number (NPT ensemble) were performed. For the expansion runs, the starting configuration was chosen to be the tessellating structure. To account for finite size effects, in each of the mixtures we simulated different system sizes, but we only report the results for 2000 particles for Mixtures 1 and 3, and for 1500 particles for Mixture 2.

The dimensionless osmotic pressure is defined as P∗ = Pl3 ,

(2.1)

where l = 1, is the length of the edge of each polyhedron simulated here, and is an arbitrary energy parameter (set to 1). In these reduced units, the approximate order-disorder transition pressure (ODP) for the different monodisperse particles are: 55, 19, 3.0, and 0.59 for tetrahedra[91], octahedra[110], cuboctahedra[87], and truncated cubes[87], respectively. The volume fraction φ of the system is just the ratio of the volume occupied by the polyhedral particles to the total volume of the simulation box.

17

Each pressure step of the expansion/compression run involved a total of 3 × 106 MC cycles (as defined below) for both equilibration and production.
Basic equilibration moves
Each MC cycle consisted on average of N translational, N rotational, N/10 flip, N/20 swap, and 2 volume move attempts. To efficiently sample phase space, the size of the move perturbations was adjusted so as to get an acceptance probability of 0.4 , 0.4 and 0.2 for the translation, rotation and volume moves, respectively. Although the size of the pressure steps was not fixed, a typical value of ∆P∗ near the phase transition was approximately 1.0, 2.0 and 0.2 for Mixtures 1,2 and 3, respectively. Flip moves attempt to rotate a chosen particle to a random orientation in the plane perpendicular to its present orientation. Swap moves involved picking randomly two particles, one of each species, and attempting to swap their positions as follows. Assume that the original positions (of the centers of mass) and orientations of these particles are r1, o1 and r2, o2, then one would attempt the changes: r1n = r2, on1 = o2 and r2n = r1, on2 = o1. Such trial moves are accepted according to the Metropolis criterion [111] (which for hard-core interactions only requires the absence of overlaps, checked via the separating axes theorem[112]). Swap moves speed-up the equilibration process, but in our systems they only succeed in isotropic phases. At pressures where the system is ordered (as obtained in trial runs), the volume moves were allowed to be triclinic [77]. All trial moves are accepted according to the Metropolis criterion [111] (which for hard-core interactions requires the absence of overlaps, checked via the separating axes theorem[112]).
18

Interfacial simulations
Interfacial NVT simulations were carried out wherein two phases, the isotropic and an ordered phase (usually the tessellating structure) were equilibrated next to each other at a constant volume and constant particle number (NVT ensemble) to test the stability of the ordered phase. Interfacial simulations take advantage of the noticeable change in φ that our systems experience around the order-disorder phase transition. This calculation entails some trial-and-error runs to determine a suitable overall system φ that lies in the two-phase region between the φ values of the isotropic and ordered phase. The cross-sectional dimensions of the interfacial simulation box were chosen from the corresponding NPT runs near the phase transition so as to accommodate an integer number of ordered-phase lattice unit cells. The box length perpendicular to the interface was chosen to be much longer than the other two, leading to larger system sizes; namely, 1768, 4500 and 4000 particles for Mixture 1, 2 and 3, respectively. For Mixture 3, we performed additional compression runs of the interfacial configuration (in an NPT ensemble) to explore in more detail the phase diagram (see details in Section 2.4.3).
Calculation of density and various order parameters
To estimate the equilibrium bulk densities of the two phases in an interfacial simulation, we used density profiles along z-axis to mark the bulk regions. For Mixture 3, in cases where the phases had a jagged interfacial boundary, we estimated the densities via a grid-based volume calculation as follows. We first isolated the region of space enclosing the phase, counted the particles therein and estimated the volume by dividing the space in a fine grid (typically spaced
19

at

a

distance

d 10

in

each

dimension,

where

d

is

the

diameter

of

the

particle’s

cir-

cumsphere) and identifying the grid-points forming the ‘interior’ of the phase

(defined

to

be

at

most

at

a

distance

d 2

away

from

the

center

of

a

particle).

To determine the orientational order in the system, we calculated orienta-

tional scatterplots where the orientation vectors of the axes of each particle are

mapped into points on the surface of a sphere (see the Supplementary Informa-

tion of [77]). In these scatterplots, a uniform distribution of points corresponds

to an absence of orientational order, while a patchy pattern gives the signature of

a particular orientational-order symmetry. We also calculated the P4 cubatic order parameter[86] to further detect cubic-like orientational order. We calculated

the Q4 and Q6 bond-order orientational parameters[113] to probe and monitor translational order. These parameters are defined as:

1

Ql =

4π 2l +

1

 

+l

2 |Q¯ lm(r)|2

−l

(2.2)

where Q¯lm(r) is given by

Q¯ lm(r)

=

1 Nb

Ylm(r)
bond s

(2.3)

where Ylm(r) are spherical harmonics for the position vector r. Although the val-

ues of these order parameters are sensitive to the crystal structure, Q6 is gener-

ically a good descriptor of crystallinity, since its value increases monotonically

with order. The value of the Q4 order parameter gives additional information about the type of crystalline structure present in the system; i.e., larger values

are associated with cubic symmetry.

To identify incipient or spatially heterogeneous ordering when (metastable) disordered phases formed in our compression simulations, we performed a cluster analysis of the configurations as follows. A set of neighboring particles

20

were assumed to form an ordered cluster if their orientations and inter-particle distances were the same as those of the tessellated structure up to some tolerance (distance d < 1.06 and cos(θ) > 0.98 where θ is the smallest angle between the axes of the two particles).

2.3.2 Polybead systems

Model particles and interactions

Polybead approximations of the perfect polyhedrons for Mixtures 1 and 2 were also implemented as a means to enact enthalpic bead-bead interactions between components and favor the formation of the expected tessellating structure. Each polybead polyhedron is built by the surface beads only, hence forming a rigid shell. We fix as 5 the number of beads per edge on any of the polyhedra (see Figure 2.3); this choice sets the degree of shape coarse-graining. Accordingly, each tetrahedron, octahedron, and cuboctahedron is made of 34, 66 and 162 beads, respectively.

Each bead is a hard sphere of diameter σ which sets the length scale of all distances given below. For the enthalpic Mixture 1, the 3 beads near the center on all facets of octahedra (type A) and on each triangular facet of cuboctahedra (type B) interact via a square-well potential:



UA,B(r)

=

 

∞ − 0

if r ≤ 1 if 1 < r < 1.5 if r > 1.5

(2.4)

This interaction results in lower (more favorable) energies when, as in the 21

Figure 2.3: Polybead models: Yellow colored beads have enthalpic interaction, while other beads on the edges have a hard-core interaction.

tessellating RCH, the triangular facets of the octahedra come face-to-face with the (triangular) facets of the octahedra. Note that for Mixture 3, one would also want to make the triangular facets of the truncated cube to be selectively attracted to the facets of the octahedra to promote the formation of the tessellating TCH. Hence, given that Mixtures 1 and 3 would conceptually be analogous in terms of the use and expected effect of selective enthalpic interactions, we only studied Mixture 1 as representative of this scenario.

In the case of enthalpic Mixture 2, it is assumed that tetrahedra and octahedra are oppositely charged in a partially screening media; i.e., like particles repel each other while unlike particles attract each other. This favors configurations where the facets of a given polyhedron type come face-to-face with facets of the other polyhedron type, just as in the tessellating ACH. We model these interactions with the following hard-core plus Yukawa potential:



Ui, j(r)

=

 

∞
e−3.5(r−1) ij r
0

if r ≤ 1 if 1 < r < 2 if r > 2

(2.5)

22

where i j = +1 for like particle-beads and i j = −1 for unlike particle beads.
In all cases, to avoid edge effects (given the ‘rounded’ shapes of our models), beads on any particle that lie on the edges only interact via hard-core interactions (in Figure 2.3 the enthalpic beads are colored in yellow). Unlike the case of Mixture 2, electrostatic interactions (as embodied by Equation 2.5 ) would be an unlikely source of the patchy selective attractions needed for Mixture 1; hence, Equation 2.4 is just a coarse model that encapsulates a generic type of short-range interaction (e.g., mediated by complementary DNA tethers or roughness[114, 115] ).
Note that polybead mixtures only approximately tessellate space (though are expected to mimic the packing of the perfect shapes) and that it is non-trivial to assign a volume to these particles. For simplicity, we use the approximate volume of the perfect polyhedron that circumscribes the polybead model, noting that this overestimates the actual volume (e.g., by adding non-existing sharp edges and vertices). Further, it is not only the edge length of the polyhedron that need not be uniform (i.e., 5) but so will be the effective edge length when considering how distinct polyhedrons try to tessellate space together around a lattice edge (these issues would be negligible in a finer model with many more beads per edge). To partially compensate for the larger effective edge lengths in the polyhedrons with smaller dihedral angles, we chose to reduce in those cases the distance between bonded nearest neighbors. For Mixture 1, bond lengths (volumes) assigned are 0.9438 (58.926) and 1.0 (245.825) for the octahedra and cuboctahedra, respectively. For Mixture 2, bond lengths (volumes) are 0.90 (20.0) and 1.0 (67.234) for the tetrahedra and octahedra, respectively. Particle volumes are only used to estimate the effective volume fractions φ of a
23

mixture.
Interfacial simulations
In the interfacial simulations, Mixture 1 comprised 500 octahedra and 500 cuboctahedra, while Mixture 2 contained 2000 tetrahedra and 1000 octahedra. The box edge along the z axis (Lz) is made ≈ 3 times longer than the other two (Lx = Ly). Simulations entailed at least 106 cycles for pre-equilibration and 106 for production, with each cycle comprising 1500 rotations, 1500 translations and one constant-volume box edge-length change. In translation and rotation moves all beads in a polybead particle move as a rigid body. In constant-volume box-edge moves we let Lz to either increase or decrease changing concurrently Lx = Ly so that the box volume remains constant; in such moves the rescaled positions of the particles centers of mass remain unchanged (as in typical volume moves). All attempted moves are accepted using the Metropolis criterion[111].
Once an overall volume fraction φ for the mixture has been identified that lies in the two-phase order-disorder region (as described in Section 2.3.1), an initial interfacial configuration is created having the desired φ value and all particles placed at their crystal lattice positions (forming a solid slab surrounded by an ‘empty’ region). This structure is then left to equilibrate so that the crystal expands and melts until reaching a stable crystal + liquid state, from which approximate coexisting φ values can be extracted. Separate simulations where the starting configuration is purely isotropic (at the same φ) were also performed to detect spontaneous nucleation and growth of the crystal phase.
24

2.4 Results for perfect polyhedrons
2.4.1 Mixture 1: Rectified Cubic honeycomb (RCH)
This is a binary mixture of cuboctahedra and octahedra in the ratio 1:1. Together, they form a cubic lattice with a 2-particle basis, very much like those formed by some binary molecules. Extensive expansion runs indicated a first-order phase transition from a solid ordered state to an isotropic state.
However, the compression runs fail to reach the tessellated state even at high pressures, at least for our relatively large system sizes adopted (trying to minimize finite-size effects) and simulation times employed. This hysteresis in the equation of state is shown in Figure 2.4 a.We also calculated different order parameters to determine if the system achieves translational or bond-orientational order. The Q4 and Q6 order parameters [113] thus calculated are shown in Figure 2.4 b, where we can pinpoint the pressure in the expansion run at which the original tessellating state loses order, while in the compression the original isotropic phase never gains any significant extent of translational order. We also calculated scatterplots (see section 2.3) to determine the orientational order and observed that both particle species lose their orientational order at the same pressure in the expansion runs. The inset of Figure 2.4 b shows one such example of scatterplots for the ordered phase before the melting transition.
It appears that the isotropic system would have to overcome a large freeenergy barrier to form the tessellating structure, if the latter is indeed the thermodynamically stable state. To further investigate this behavior, we performed NVT ensemble interfacial simulations at volume fractions close to the phase
25

(a) (b)
(c) (d)
Figure 2.4: Results for Mixture 1 (RCH): a) Equation of State for the compression and expansion MC runs. Coexistence pressure from interfacial simulation has been marked as Ptrans. ODPA and ODPB are the ODPs for the pure octahedra and cuboctahedra, respectively. b) Plot of Q4 and Q6 as a function of pressure in compression and expansion MC runs. The scatterplots for the orientational order for each of the particle species at P∗ ≈ 5.9 in the expansion run are given in the inset. c) Simulation snapshot of glassy disordered state from compression run at P∗ = 14.9, φ ≈ 0.61. d) Snapshot from the interfacial NVT simulation at φ= 0.54, and P∗ ≈ 5.9. Purple = Cuboctahedra, Green = Octahedra. The adjoining plot shows the density profile along the z-axis, where the green curve is the actual density and the blue lines show the estimated bulk density of the two phases.
transition (as observed in the expansion run), starting from the ordered state with the appropriate box size. These simulations show that the ordered phase is stable and coexists with the isotropic phase. This coexistence is stable within a
26

range of composite system densities ranging from 0.54 to 0.59 (above which the tessellating structure outcompetes the isotropic phase). The fact that one needs to seed the ordered phase for it to stay or grow suggests again that a large-free energy barrier exists for the self-assembly of this tessellating phase. This barrier arises from the numerous cooperative particle rearrangements required to form a critical-sized nucleus of the ordered phase from a dense isotropic state.
2.4.2 Mixture 2: Alternated Cubic Honeycomb (ACH)
This is a mixture of equal-edge-length tetrahedra and octahedra in the ratio 2:1. Similar to Mixture 1, we observe that our MC compression runs fail to reach the tessellating ordered state. The dense system thus formed lacks positional as well as orientational order, which is evident in the plots of the order parameters for both the compression and the expansion runs. The equation of state as shown in Figure 2.5 a, clearly show two branches in the high pressure regime, one for the ordered tessellating structure from the expansion run, and the other for the low-density, glassy, disordered structure from the compression run. Similar to Mixture 1, we observe that both particle species lose orientational order at the same pressure as is seen in the scatterplot in the inset of Figure 2.5 b. This shows that both species exhibit a tight coupling in their tendencies for orientational ordering.
Interfacial NVT simulations at volume fractions close to the isotropic-crystal phase transition show that the ordered and disordered phases maintain a stable coexistence for 0.52 < φ < 0.6. Thus, we see similar qualitative and quantitative behavior in Mixtures 1 and 2 in that, although the tessellated structure is found
27

(a) (b)
(c) (d)
Figure 2.5: Results for Mixture 2 (ACH): a) Equation of State for compression and expansion MC runs as a function of volume fraction φ. ODPA and ODPB are the ODPs for the pure tetrahedra and octahedra, respectively. b) Plot of Q4 and Q6 as a function of pressure in compression and expansion MC runs. The scatterplots for the orientational order for each of the particle species at P∗ ≈ 4.62 in the expansion run are given in the inset. c) Simulation snapshot of glassy disordered state from compression run at P∗ = 136, φ ≈ 0.68. d) Snapshot from the interfacial NVT simulation at φ= 0.54, and P∗ ≈ 37. Green = Octahedra, Orange = Tetrahedra. The adjoining plot shows the density profile along the z-axis, where the green curve is the actual density and the blue lines show the estimated bulk density of the two phases.
to be stable from the interfacial simulations, the nucleation of such a structure from the isotropic state in compression runs seems to be hindered by a large free-energy barrier.
28

2.4.3 Mixture 3: Truncated Cubic Honeycomb (TCH)
Compared to Mixtures 1 and 2, Mixture 3 is observed to have a very different behavior upon compression: Instead of attaining a glassy, disordered state, the system undergoes phase separation into isotropic and ordered phases. The ordered phase is rich in the bigger truncated cubes, which readily order into a cubic lattice (see Figure 2.6).
Interestingly, even the expansion runs show phase separation in a small region before melting into a completely disordered isotropic state. To probe the thermodynamic behavior, we first performed interfacial simulations in the NVT ensemble at φ ≈ 0.54. This simulation showed the coexistence between a truncated cube-rich ordered phase and an octahedron-rich isotropic phase starting at a pressure of P∗trans1 = 0.9. We then gradually compressed this configuration in an NPT ensemble. The simulations were set to allow for octahedrons to attempt large translations to circumvent slow diffusion and potentially find near or distant open pockets in the ordered region. Using the stoichiometric (equimolar) composition, we found that the octahedra-rich phase tended to become too small to resolve its bulk properties (like density and composition). To overcome this, we used a lower global composition of truncated cubes (xB = 0.25), which should not affect the intensive properties of the coexistence phases for states in the two-phase region (it only affects the relative amounts of each phase according to the lever rule). Results from these simulations (see compositions in the inset of Fig. 2.6 b and snapshots in Fig. 2.6 c and d) show that for P∗ > 2 the truncated cube-rich phase approaches the stoichiometric composition (xB = 0.5); however, the structure of this phase does not seem to approach that of the tessellating compound but rather it contains small interstitial pockets in between
29

(a) (b)
(c) (d)
Figure 2.6: Results for Mixture 3: a) Proposed Thermodynamic equation of state from expansion, compression and interfacial runs. The branch corresponding to the truncated cube-rich phase will likely become metastable for some P∗ > 3. ODPB is the ODP for pure truncated cubes. b) Plot of P4-order parameter for each component in the different phases; squares correspond to truncated cubes and diamonds to octahedra in the respective phases. The inset shows a plot of composition (expressed here in terms of xB = mole fraction of truncated cubes) vs. Pressure in the two-phase region . Snapshots from interfacial simulations showing coexisting truncated cube-rich cubic phase and octahedral-rich isotropic phase: c) Configuration at P∗ ≈ 1.1; d) Configuration at P∗ ≈ 2.3. Green = Truncated Cubes, Red = Octahedra. These snapshots are for a 4096-particle system with 25% Truncated Cubes for better definition of the bulk regions. The adjoining plots show the density profile along the z-axis, where the green curve is the actual density and the blue lines show the estimated bulk density of the two phases.
the truncated cubes where more than one octahedron often reside. In such a remarkable structure, that we will refer to as cubic semi-crystal given that the octahedra largely exist in positional and orientational disorder (as seen in Fig. 2.6
30

b-d), must possess a higher entropy than the tessellating compound (where the octahedra would have positional and orientational order) for some intermediate range of packing fraction. It is therefore conjectured that this semi-crystal phase would transition into the tessellating structure at higher pressures but this is precluded by the lack of ergodicity in our simulations. From the expansion runs started from the tessellated compound structure (with xB = 0.5), we can detect the onset of a phase transition at P∗trans2 ≈ 2.7 below which the system phaseseparates into the cubic semi-crystal phase (with xB > 0.5) and octahedron-rich phase found before from the interfacial simulations. While only free-energy calculations could decide when the tessellated compound may outcompete the semi-crystal (with xB = 0.5), we observe that for P∗ > 3 the former packs more densely than the latter (for the same P∗); this more efficient packing may translate into a higher packing entropy and a concomitant higher stability. Note also that P∗trans2 is much lower than the ODP = 19 of the pure octahedra. Combining all these results, we piece together the approximate equilibrium equation of state shown in Figure 2.6 a.
Mixture 3 clearly shows a far richer phase behavior than Mixtures 1 and 2 since at least one distinct two-phase region is observed between the completely isotropic and the tessellated compound phases. Part of this difference can be ascribed to the large asymmetry in the ratio between component volumes (≈ 29) in Mixture 3 and that the truncated cubes adopt the same crystalline order in both its pure crystal and the tessellating compound. We hence conjecture that at high pressures octahedrons and truncated cubes should form a solid solution for compositions of the latter xB > 0.5, as illustrated in Fig. 2.1 d. Indeed, the phase behavior of Mixture 3 can be recapitulated by following the vertical red line in Figure 2.1 c going upward and taking A = octahedra and B = truncated
31

cubes: as the isotropic phase is compressed it will encounter first a two-phase region where phases I and AB(ss) coexist before the AB(c) compound would ensue at high pressures.
2.4.4 Analysis
For Mixtures 1 and 2, interfacial simulation results provide support to the tenet that the isotropic phase will become unstable relative to the putative ordered tessellating phase at conditions (pressure or density) above those where their coexistence is observed. Interfacial simulations cannot resolve the question of whether alternative ordered structures could be more stable; however, the high packing efficiency (and associated translational entropy) of the tessellating structure(s) makes any alternative non-tessellating crystal highly unlikely as a stable phase at high densities. Our results suggest that, in purely entropic (hardcore) particles, there exists a significant entropic barrier to form the tessellating phase from the homogeneous nucleation out of an isotropic phase. This justifies the exploration of strategies to bias the kinetics of the process toward the tessellating structure; e.g., via face-specific attractions [107] as discussed in Section 2.5. The formation of tessellating structures appears to start via nucleation of small pockets of the ordered structure; however, these pockets fail to grow or merge in to give a macroscopic order to the system. To study this further, we performed a cluster analysis (see section 2.3) of the configurations obtained in compression runs that lead to glassy disordered system. The result of such an analysis is shown in Figure 2.7 and typical clusters are depicted in Figure 2.8. As shown in Figure 2.7, the size of the largest cluster shows a sharp increase just after the transition pressure, signaling a clear tendency toward local
32

(a) (b)
Figure 2.7: Results of the cluster analysis: Volume fraction and size of the largest cluster size as a function of the pressure P∗ from compression runs for a) Mixture 1 (RCH), and b) Mixture 2 (ACH). The insets show plots of the mean square displacement (MSD), in units of polyhedral edge length squared, after 40000 MC cycles at pressures around P∗trans.
(a) (b)
Figure 2.8: Sample snapshots of ordered clusters in a) Mixture 1 (RCH) and b) Mixture 2 (ACH). Purple = Cuboctahedra, Green = Octahedra, Orange = Tetrahedra. ordering. However, these clusters fail to grow into a single tessellating structure, likely hindered by the slower particle dynamics associated with denser states. We probed this slowing down by computing the mean square displacement (MSD) after 40000 MC cycles using structures generated in the compression runs. These MSD results, shown in the insets of Figure 2.7, indicate that
33

Table 2.1: Order-disorder transition pressures (ODPs)

Pure species Mixtures with
Octahedra

Octahedra 19

Cuboctahedra 3.0 5.9
(Mixture 1)

Tetrahedra 55 34
(Mixture 2)

Truncated Cubes 0.59 0.9
(Mixture 3)

around P∗trans the translational mobility decays quickly but does not become negligible. The fact that the MSD for the two components have similar values (for a given P∗) suggests that their motions tend to be correlated. However, forming larger crystallites may depend more crucially on rare concerted reorientations of multiple particles (or small ordered clusters) than on concerted particle translations. A detailed study of the kinetics of the order-disorder phase transition, and of the intervening free-energy barrier will be the object of our future efforts to complement the results of this work and provide further insights into the actual self-assembly and nucleation mechanism.

The results for Mixture 3 suggest that skewed volume ratios in the components can help reduce the barrier for the ordered assembly of the larger particles if the ordering structure of the monodisperse case is maintained in the tessellating compound. However, Mixture 3 also illustrates that very high size ratios will induce demixing at intermediate pressures, making it kinetically and entropically difficult for the smaller particles (residing in a separate phase) to remix at higher pressures and find their ideal positions in the tessellating structure. In our simulations, such diffusion limitation was only partially alleviated by the non-physical nature of the MC moves. A disparity in component particle size will translate into a disparity on ODPs of the pure components, which not only sets the bounds for the ODP of the mixture (with the stoichiometric composition) as summarized in Table 2.1, but also modulates different two-phase regions in the entire phase diagram as illustrated in Figure 2.1. More generally,

34

it can be conjectured that optimal intermediate volume ratios for the particles could be found such that demixing is avoided and a more facile self-assembly into tessellating compounds is attained.
2.5 Results with Polybead Models
Figure 2.9 a shows the phase diagram for Mixture 1 (from interfacial simulations), describing the effect of changing the strength of the attraction (squarewell depth) on the coexistence volume fractions. The results for = 0 correspond to the hard-core case which would be comparable to those found for the perfect shapes; note however that the coexistence values of φ do not exactly agree with those reported in Figure 2.4 a given the differences in the models and the arbitrariness in defining the polybead particle volumes. As | | is increased, the two-phase region (or the miscibility gap) becomes wider with the liquid and crystal concentrations becoming more disparate. This happens because as the favorable inter-species attraction becomes stronger, the tessellating crystal becomes denser and gains stability relative to a dense isotropic liquid. The results in Figure 2.9 a correspond to a starting crystal-slab configuration where the overall volume fraction was fixed at φ = 0.61 for ≤ 0.3 and at φ = 0.554 for > 0.3; using a different global φ simply changes the relative amounts of the two phases in the interfacial box (according to the ‘lever-rule’) leaving the coexisting φ’s of the two phases unaffected. Figure 2.9 b (top) shows a sample configuration of the interfacial system. Using simulations that start from a uniform isotropic state did not always spontaneously nucleate the crystal and produce a crystal-liquid two-phase state, at least in runs of 2 × 106 cycles and for the overall φ values used. Spontaneous nucleation only happened for ≥ 0.6
35

(a) (b)
Figure 2.9: a) Phase diagram for polybead model of Mixture 1 for varying strengths of the selective square-well attraction between triangular facets of cuboctahedra and octahedra. The case = 0 corresponds to the athermal mixture. Errorbars are approximately 3 times the width of data symbols. (b) Sample configurations from interfacial simulations for the polybead enthalpic model for Mixture 1 at = 0.7 and overall volume fraction φ = 0.554. Top: Starting from a pre-ordered slab; bottom: starting from a uniform isotropic state (the orientation of the crystal region in the latter is different). Purple = cuboctahedra, green = octahedra)
and produced two-phase states whose coexisting structures and φ’s are consistent with those reported in Figure 2.4 a. A sample result for = 0.7 is shown in Figure 2.9 b (bottom). Interestingly, the spontaneous two-phase states found for
= 1 had many more defects that those found for = 0.65 (for example) after 106 cycles; in the former case it appears that independent stable seeds formed early had more difficulty coalescing, hence leading to grain boundaries. While much longer simulations tend to anneal those defects, our results suggest that there could be a range of values which is kinetically optimal in the sense of allowing the formation of the crystal compound faster and with minimal defects: a too small is insufficient to timely overcome the free-energy barrier to nucleation of the crystal seed, while a too large may lead to a spinodal-like crystallization where multiple crystal seeds tend to get stuck in their initial orientations. A quantitative determination of such optimal conditions would necessitate a sys-
36

tematic study (for different ) of the effect of varying degrees of supersaturation on nucleation kinetics; such a study lies beyond the scope of this work.
(a) (b)
Figure 2.10: a) Phase diagram for polybead model of Mixture 2 for varying strengths of the selective contact energy value of the Yukawa potential between the facets of tetrahedra and octahedra. The case = 0 corresponds to the athermal mixture. Errorbars are approximately 3 times the width of data symbols. (b) Sample configurations from interfacial simulations for the polybead enthalpic model for Mixture 2 at = 0.5 and overall volume fraction φ = 0.606. Top: Starting from a pre-ordered slab; bottom: starting from a uniform isotropic state. Green = octahedra, Orange = tetrahedra)
Figure 2.10 a shows the phase diagram for Mixture 2 (from interfacial simulations), describing the effect of changing the strength of the Yukawa attraction/repulsion contact energy on the coexistence volume fractions. We observe similar trends as those described for Mixture 1 in the previous paragraphs and similar comments apply here as well. We see in Figure 2.10a that the broadening of the two-phase region gap seems more gradual than in Figure 2.9a but, of course, the parameters in these figures are not comparable since they refer to different interaction potential models. The results in Figure 2.10a correspond to a starting crystal-slab configuration where the overall volume fraction was fixed at φ = 0.61 for ≤ 0.5 and to φ = 0.574 for > 0.5.
37

In simulations that started from a homogenous isotropic phase, spontaneous formation of a crystalline region was observed for ≥ 0.5 (during 2 × 106 cycles). Compared to Mixture 1, in this case ordered domains seemed to take much longer times to anneal local defects to form a large slab of the crystal. This difficulty was more pronounced for large | |. Additional simulations were conducted for the case where only attraction between unlike particles species are enacted; i.e., by setting i j = 0 for like beads (which then only exhibit hard-core interactions). These led to a phase diagram (not shown) that was almost indistinguishable from that shown in 2.10 a. This indicates that (for our choice of model parameters) it is primarily the attractive interactions that determine the isotropic-crystal phase boundaries.
While we did not simulate an enthalpic polybead model for Mixture 3, one can readily anticipate what will happen if we were to use a similar type of preferential attraction between interspecies triangular facets as that implemented for Mixture 1: we can tilt the balance toward the TCH compound sufficiently to eliminate the two-phase region from the equation of state so that the TCH can be spontaneously nucleated upon compression of the isotropic phase.
2.6 Conclusions
As an extension to the study of the phase behavior of single-component polyhedral particles, we studied here the self-assembly of a set of binary mixtures which can tessellate space. Since in these cases the particles individually cannot tessellate space, the tessellated structure provides a strong candidate for a thermodynamically stable phase at high enough pressures. While interfacial simu-
38

Figure 2.11: Sketch of the free energy landscape as the binary blend is compressed from a state where the isotropic liquid is stable (red curve) to one where other phases are stable (blue curve). In the latter situation, a system started in the isotropic basin may either get kinetically trapped therein, transition into the two-phase basin, or transition into the crystal compound basin. In the athermal Mixtures 1 and 2, ∆F1∗ and ∆F2∗ in the solid blue curve are too large to timely escape the isotropic basin; in the athermal Mixture 3, ∆F1∗ is not only smaller than ∆F2∗ but also small enough that the two-phase basin is rapidly accessed. The selective enthalpic interactions used in the polybead mixtures not only deepen the crystalline compound basin but also reduce ∆F2∗, as depicted by the dashed green curve.
lations confirm that the tessellated structure is the stable phase at high densities
for two of those mixtures, large hysteresis is observed in the equations of state
obtained from compression and expansion runs.
Figure 2.11 provides a conceptual summary of the analysis of the results with
our three mixtures and for both athermal and enthalpic systems. In reference to
39

Figure 2.1, we consider the changes in the free-energy landscape of a system with the compound stoichiometric composition as it traverses the isotropiccrystal transition (i.e., going up on the red line in Figure 2.1 b for Mixtures 1 and 2, or Figure 2.1 d for Mixture 3). We hypothesize that there exist at least three competing basins corresponding to the isotropic (or amorphous) state, the crystalline compound, and a state with two separate phases (e.g., a crystal and a liquid). Immediately above the pressure (or global volume fraction) at which the isotropic phase becomes metastable, the deepest basin would correspond to the crystalline compound for Mixtures 1 and 2, and to the two-phase state for Mixture 3. Upon compression of an isotropic mixture into its metastable region (blue curve in Figure 2.11), the rate at which the system would transition into the single compound crystal basin or the two-phase separated basin will primarily depend on the relative heights of the free energy barriers ∆F1∗ and ∆F2∗. If ∆F1∗ >> kT and ∆F2∗ >> kT , then the system may simply be kinetically trapped in a metastable disordered state over the length of a typical (unbiased) simulation run; this is what athermal Mixtures 1 and 2 seem to experience. If ∆F1∗ < ∆F2∗ and ∆F1∗ ≈ O(kT ), then the separated two-phase state will be observed in simulation, as in the case of the athermal Mixture 3. Conversely, if ∆F2∗ < ∆F1∗ and ∆F2∗ ≈ O(kT ), then the crystal compound will be spontaneously observed in a simulation, as in the case of the polybead thermal Mixtures 1 and 2 (for sufficiently large interaction strengths). Of course, the selective enthalpic interactions used in the polybead mixtures not only reduce the height of ∆F2∗ but also deepen the depth of crystalline compound basin to further stabilize that phase.
Note that the analysis above is strictly qualitative but a detailed quantitative study is workable to obtain free energy barriers (e.g., via Umbrella Sampling
40

methods [116–119]) and transition pathways (e.g., via forward flux sampling methods [117, 119, 120]). In such methods, the choice of the order parameter used to track the transition will be critical to the computational efficiency and the accuracy of the results. If the disorder-to-order free-energy barriers do turn out to be very large for Mixtures 1 and 2, this could make these systems very appealing as models of purely entropic glass-formers (as some binary mixtures of hard spheres [121] ). Since many polymeric glass-formers tend to be fragile, non-polymeric models of glass-formers are of great interest, both in experiments and in fundamental studies of glass transitions [121–123].
Formulating ‘ideal’ multicomponent systems which minimize the disorderto-order free energy barrier is an interesting but very challenging design question. As described earlier, the volume ratio between components is one property worth fine tuning, especially if the bigger particles can easily form a ‘skeleton’ for the assembly of the other component. Engineering face-specific attractions between the colloidal particles is another practical way of reducing the alluded free-energy barrier. The presence of mesophases or external interfaces could help speed up such an assembly, similar to the effect of an intermediate in a catalyst-assisted reaction or of heterogeneous nucleation.
From our simulation results, design rules and guidelines are only beginning to emerge to formulate nanoparticle polyhedral compounds that assemble into a desired superstructure. Apart from relative shapes and sizes, phase behavior of individual particles is critical to predict stable and metastable states of the mixtures as outlined in Figure 2.1. Overall, these results take a step towards a ‘chemistry’ of such nanoparticles that should add to our arsenal of strategies for targeted material design.
41

CHAPTER 3 GENERIC BINARY MIXTURES
3.1 Introduction
Shape1 anisotropy of colloidal nanoparticles has emerged as an important design variable for engineering assemblies with targeted structure and properties. In particular, a number of polyhedral nanoparticles have been shown to exhibit a rich phase behavior[77]. Since real synthesized particles have polydispersity not only in size but also in shape, we explore here the phase behavior of binary mixtures of hard convex polyhedra having similar sizes but different shapes. Choosing representative particle shapes from those readily synthesizable, we study in particular four mixtures: (i) cubes and spheres (with spheres providing a non-polyhedral reference case), (ii) cubes and truncated octahedra, (iii) cubes and cuboctahedra, and (iv) cuboctahedra and truncated octahedra. The phase behavior of such mixtures is dependent on the interplay of mixing and packing entropy, which can give rise to miscible or phase-separated states. The extent of mixing of two such particle types is expected to depend on the degree of shape similarity, relative sizes, composition, and compatibility of the crystal structures formed by the pure components. While expectedly the binary systems studied exhibit phase separation at high pressures due to the incompatible pure-component crystal structures, our study shows that the essential qualitative trends in miscibility and phase separation can be correlated to properties of the pure components, such as the relative values of the order-disorder transition pressure (ODP) of each component. Specifically, if for a mixture A+B we have
1This chapter is reproduced from reference [124] : ‘Phase behavior of binary mixtures of hard convex polyhedra’, with permission from the Royal Society of Chemistry.
42

that ODPB <ODPA and ∆ ODP = ODPA - ODPB, then at any particular pressure where phase separation occurs, the larger the ∆ ODP the lower the solubility of A in the B-rich ordered phase and the higher the solubility of B in the A-rich ordered phase.
Interest in material design based on nanocrystal assemblies has been rapidly increasing over the past decade. The choice of building blocks, size and shape, composition and surface functionalization offers multiple avenues to taylor the properties of these assemblies. Going beyond the prototypical spherical nanoparticles, researchers have explored the effect of shape anisotropy on colloidal self-assembly afforded by the synthesis of cubes, rods, plates, disks and particles with core-shell structure [28][125][37][30]. With the advent of new particle synthesis methods[126][127][108][128] polyhedral nanoparticles are becoming ubiquitous in providing varying extent of shape ‘anisotropy’. Suspensions of a range of polyhedral shapes have been shown to produce interesting partially ordered ‘mesophases’ [77] [90][91][86] at intermediate volume fractions. The mixing of nanoparticles of different chemical composition could provide a simple way to prepare ordered structures with more desirable properties; e.g., if a dopant component adds some functionality to the host component in a matrix. If the particles differ not only in composition but also in shape, one may gain additional control over the extent of infiltration by the dopant component, how their particles orient and what spatial environment they encounter.
43

Figure 3.1: Figure showing some of the polyhedra synthesized at different steps of a modified polyol process. The arrow shows the order of the shapes produced as the reaction is allowed for longer times.
3.2 Polyol Process and choice of systems studied
A range of different polyhedral nanocrystals have become readily synthesizable by controlling the growth step in a modified polyol process for the formation of nanocrystals made of gold [38][39] and other inorganic materials[63][64] ; Figure 3.1 shows some of the particle shapes typically produced this way. This approach also readily leads to mixtures of these polyhedra (shape bidispersity) by stopping the growth process at different stages. The phase behavior of such binary mixtures (of particles having similar size but different shape) is expected to depend on multiple factors, including the phase behavior of the individual polyhedra and in particular their order-disorder transition pressure (ODP) and the crystal lattice they assemble into. A specific class of tessellating mixtures[129] has been studied before; in that case the pure components exhibit incompatible crystal lattices, but at a precise (stoichiometric) composition and size ratio, they can assemble into binary crystalline compounds. However, most mixtures of polyhedra with incompatible crystal lattices (for the monodisperse
44

systems) would be expected to phase separate at higher volume fractions into ordered phases resembling those of the pure components.
When a mixture of two polyhedral shapes A + B does phase separate, it is important to explore the extent to which they mix in the different phases; i.e., how much B is incorporated in the A-rich phase and how much A is incorporated in the B-rich phase. Such information will be useful, e.g., in predicting the limiting compositions at which such bidisperse mixtures can exist without phase separation. Two components that individually form the same type of mesophase may give rise to mixtures that exhibit a similar mesophase. For example, appreciable mixing of components has been observed in the case of polydisperse mixtures of rigid rods of different length which form a nematic phase[130], and of cubes of different size that form a cubatic mesophase[87].
With these considerations in mind, we study here the self-assembly of binary mixtures of convex polyehdra. As a reference case, we study first the cube + sphere mixture (CS), taking the sphere as the infinite-facet limit of a regular polyhedron. Further, we choose three representative binary mixtures, each made from polyhedra that have been synthesized by various experimental methods [38][39][68] : cubes + truncated octahedra (CTO), cuboctahedra +cubes (COC), and cuboctahedra + truncated octahedra (COTO). These mixtures represent different degrees of similarity in terms of rotational symmetry, asphericity [77] and the crystal lattice they individually assemble into. The presence of similar mesophases[77] may also affect this degree of mixing. Large difference in ODPs between components is associated with larger difference in sizes and hence is expected to result in more ‘incompatible’ mixtures, that readily phase separate. Conversely, components with similar ODPs and similar mesophases
45

could be expected to better mix with each other. Our aim in this paper is to determine approximate pressure-composition phase diagrams for each of these mixtures to try to elucidate the relation between the phase behavior of the mixture and that of the pure components. Although there are many experimental methods prevalent in the literature that allow synthesis of concave shapes, as a first step, we aim to keep our analysis simple by restricting ourselves to more common convex shapes.
The CS system represents a limiting case when one of the components (spheres) has no anisotropy. In this case, the individual shapes and their crystal structures are quite incompatible and they are expected to phase separate above the components’ ODPs. In our simulations, we have assumed that the side of the cube is equal to the diameter of the sphere. While such a choice does fulfill the criterion of having components similar in size, it will necessarily affect the precise location of phase boundaries in the phase diagram and also create an asymmetry in the relative diffusivities of the components in a given phase (with the smaller component typically having larger mobility).
The other mixtures studied, namely, CTO, COC and COTO represent cases of components with varying degrees of similarity in shape and size. The CTO mixture corresponds to components which are rather distant in the polyol process (see Figure 3.1); this mixture also embodies a peculiar feature where the two individual components are space tessellating (at close packing) but a mixture of any composition is not. Such a feature and the fact that their ordered phases have very different symmetry suggest that the CTO system should be largely incompatible (not unlike the CS system). Considering that cubes and truncated cubes have almost identical phase behavior and very similar shapes, one can
46

Figure 3.2: Snapshots showing the 4 mixtures studied: a) CS mixture (cubes+ spheres), b) CTO mixture (cubes+ truncated octehedra), c) COC mixture (cubes + cuboctahedra), d) COTO mixture (cuboctahedra + truncated octahedra). The relative sizes accurately describe the size ratios in simulations.
consider the mixture of cubes and cuboctahedra (COC) as representing shapes which are essentially neighbors in the polyol process (see Figure 3.1). Further, the partial similarity of the ordered structures of the pure components suggest that the COC mixture could have an intermediate degree of compatibility. The mixture of cuboctahedra and truncated octahedral (COTO) can be considered as potentially the most compatible given that the components are proximal in the polyol process (Fig. 1) and that both form a rotator mesophase. In each of the mixtures, the size ratios of the components (see next Section) are consistent with those obtained in a typical polyol process[38][39].
47

3.3 Methodology

In our simulations, we assume the particles to be interacting via hard-core potentials, which amounts to ensuring that particles never overlap. Extensive expansion and compression Monte Carlo (MC) runs were performed to map out the equation of state of each of the mixtures, at constant pressure and particle number (NPT ensemble). Although we observe hysteresis between expansion and compression runs, we use expansion runs to estimate the transition pressures because they typically need to surmount a smaller free energy barrier at the transition points and are hence expected to more closely follow thermodynamic behavior. Unless otherwise indicated, the mixtures were simulated at the same composition of 50 %. We used a total of 2560 particles for CS, CTO and COTO mixtures and 2048 particles for COC mixture. As indicated earlier, the size of the components was chosen based on typical results of the modified polyol process. For such a process, Seo et al. [38] report the average edge lengths of different shapes to be 98, 120 and 145 nm for truncated octahedra, cuboctahedra and cubes respectively. Of course, synthesis conditions can be altered so that a given particle shape can be obtained with different sizes, but choosing sizes consistent with those attainable within a single growth experiment underscores a simple possible strategy to obtain the mixtures studied in this work.

The dimensionless osmotic pressure is defined as P∗ = Pl3 ,

(3.1)

where l = 1, is the characteristic length in each of the mixtures simulated here. In mixtures involving the cubic particles, this characteristic length is the length of the side of the cube, while in COTO mixture, it is the side of the imaginary cube from which the cuboctahedron is cut. is an arbitrary energy parameter

48

(set to 1). In these reduced units, the edge-lengths, particle volumes, approximate order-disorder transition pressure (ODP) and mesophase-crystal transition pressure (MCP) for all the particles studied are shown in Table A.1. The volume fraction φ of the system is just the ratio of the volume occupied by the polyhedral particles to the total volume of the simulation box.

Each pressure step of the expansion/compression run involved a total of

3×106 MC cycles (as defined below) for both equilibration and production. Each

Table 3.1: Reference data for the pure components studied. ODP describes order-disorder transition pressure which in most cases is an isotropicmesophase transition. MCP denotes the mesophase-crystal transition pressure. The references used for the data are listed alongside the shape.

Shape Cube[77] Cuboctahedron[87] Sphere[131] Truncated Octahedron[77]

Edge-length 1 0.707 1 (diameter) 0.354

Volume 1 0.833 0.524 0.5

ODP 6.3 7.1 11.5 14

MCP 8 14 28

MC cycle consisted on average of N translational, N rotational, N/10 flip, N/20 swap and 2 volume move attempts. Flip moves attempt to rotate a chosen particle to a random orientation in the plane perpendicular to its present orientation. Swap moves involved picking randomly two particles, one of each species, and attempting to swap their positions as well as orientations (since such moves are more likely to succeed with swapped orientations). Swap moves are essential to speed-up the equilibration process by allowing particles to move arbitrarily far from their original positions circumventing the slow diffusion associated with dense phases. Swap moves, however, can only be effective if the particle shape and size of the two components is similar and densities are moderate to allow some wiggle room around a lattice site to accommodate different particle shapes. At pressures where the system is ordered (as obtained in trial runs), the volume moves were allowed to be triclinic [77]. The size of the move perturba-

49

tions was adjusted so as to get an acceptance probability of 0.4, 0.4 and 0.2 for the translation, rotation and volume moves, respectively. Although the size of the pressure steps was not fixed, a typical value of ∆P∗ near the phase transition was approximately 0.8, 1.3, 0.8 and 1.6 for CS, CTO, COC and COTO mixtures respectively. All trial moves are accepted according to the Metropolis criterion [111] (which for hard-core interactions requires the absence of overlaps, checked via the separating axes theorem[132] for any two polyhedra or Arvo’s algorithm for the cube-sphere case [133]).
Interfacial simulations where two phases and the intervening interfaces are present in the same box were used to estimate coexistence conditions. To facilitate the formation of distinct bulk regions, the box was at least four times more elongated along one direction that the others, so that the interfaces would form perpendicular to that axis. Once interfaces form, the longitudinal pressure (i.e., the one acting on a plane parallel to the interfaces) provides the proper estimation of the coexistence pressure since the transverse pressure contains a contribution from the surface tension [134]. Thus, in such cases we chose the transverse box length (which sets the dimensions of the interface) to be commensurate with the lattice spacing for the individual phases at the given pressure, only sparingly allowing changes in transverse dimensions to help relieve any build-up of stresses. Special care must be taken when dealing with interfacial simulations involving two solid phases where the box cross section (perpendicular to the long axes) must be chosen so that it can properly accommodate the unit cells of the distinct crystal lattices. Unfortunately, this is hard to achieve over a wide range of pressures. Hence, although triclinic volume moves did allow for box deformations that can rotate the lattices to alleviate internal stresses, our results in such regions are expected to have larger errors than at lower pres-
50

sures.

Unless otherwise indicated, interfacial simulations were performed for an equimolar global composition as it was expected that, if phase separation occurs, the compositions of the coexisting phases would be relatively symmetric (on account of the particle-size similarity) and hence lead to similar amounts of the two phases. To estimate the equilibrium bulk densities of the two phases in an interfacial simulation, we used density profiles along the z-axis to mark the bulk regions.

We calculated the Q4 and Q6 bond-order orientational parameters [113] to

probe and monitor translational order. These parameters are defined as:

1

Ql =

4π 2l +

1

 

+l

2 |Q¯ lm(r)|2

−l

(3.2)

where Q¯lm(r) is given by

Q¯ lm(r)

=

1 Nb

Ylm(r)
bond s

(3.3)

where Ylm(r) are spherical harmonics for the position vector r. Although the val-

ues of these order parameters are sensitive to the crystal structure, Q6 is gener-

ically a good descriptor of crystallinity, since its value increases monotonically

with order. The value of the Q4 order parameter gives additional information

about the type of crystalline structure present in the system; i.e., larger values

are associated with cubic symmetry.

To determine the orientational order in the system, we calculated the P4 cubatic order parameter[86] which gives information about cubic-like orientational order. To mark the boundary between the rotator mesophase and the crystalline phase, we used a combination of the P4 order parameter and the orientational scatterplots. Since such a transition is continuous, we set the up-

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Figure 3.3: Representative orientational scatterplots used for determining the orientational order. From top a) Truncated Octahedra (TO) in CTO mixture, b) cuboctahedra (CO) in COC mixture, c) CO in COTO mixture, d) TO in COTO mixture. For pressures above the phase separation, the scatterplots show the orientational order for the particles in the ordered phase that is rich in the named component only.
per limiting P4 value of the rotationally disordered phase to be 0.3 (note that the maximum value of P4 for perfect cubic order is 0.583). The scatterplots were generated by plotting the orientational axes of each the particles on a unit sphere. In these scatterplots, a mostly uniform distribution of points signals the absence of orientational order, which along with the presence of translational order (estimated through the Q6 order parameter) identifies a rotator mesophase. A patchy pattern gives the signature of a particular orientational-order symmetry, which is absent in a rotator mesophase. A set of representative scatterplots for the systems studied here are shown in Figure 3.3.
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3.4 Results
3.4.1 Cubes + Spheres (CS) Mixture
As previously indicated, this system represents a base case of components with widely different shapes and monodisperse phase behavior. Cubes exhibit a mesophase[135] for a small range of volume fractions (0.51-0.54) which separates isotropic and cubic crystalline phases[77]. There is disagreement in the literature [88][136] about the precise nature of such a mesophase, which hinges on the criterion adopted for classifying a phase as being crystal-like or liquidcrystal-like. For the range of volume fractions specified above, while the average particle positions are crystal-like (despite a high vacancy content), the variance of the position fluctuations and particle mobilities are liquid-like [87]. Since the current taxonomy is not completely satisfactory, for concreteness we will henceforth refer to this mesophase as cubatic. Spheres on the other hand, show a single phase transition from isotropic to FCC crystal. While spheres have full rotational symmetry, cubes do not and possess a relatively large asphericity γ (= ratio of circumradius to inradius = 1.732). In our simulations, the side of cubic particle was set to be equal to the diameter of the spherical particle.
Extensive MC runs for the equimolar mixture exhibit an isotropic, fully mixed phase, up to P* ≈ 8, close to the MCP for cubes. Thus as phase separation ensues above P* ≈ 8, the cubes are already in a crystalline state, while spheres are still disordered. At a pressure P* ≈ 12, the spheres order, a value which is only slightly higher than the ODP for pure spheres (P∗ ≈ 11.54). Note that generally, for any A+B mixture, the apparent ODP in the A-rich phase is expected to be near but slightly above the ODP of the pure A system because of
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Figure 3.4: Summary of the results for the CS mixture. a) Equation of state with ODPs for the individual particles shown for reference. Here ODPA is the ODP of spheres while ODPB is the ODP of cubes. b) Plot of Q4 and Q6 order parameters. c) Pressure vs. composition phase diagram with different regions marked with the order of the constituent phases. Here xB is the mole fraction of cubes. d) Snapshots from the interfacial simulations of 3 representative regimes of the phase diagram (spheres are cyan and cubes are dark red). Bottom: cubic+isotropic phases at P* ≈ 8. Middle: Same phases at P* ≈ 11. Top: Cubic + FCC phases at P* ≈ 15.
the disordering effect of the B particles present; our observations are consistent with this expectation.
We demarcate the putative cubatic region by a combination of increased P4 order parameter and high translational mobility. We estimate the latter by calculating the mean square displacement (in a pseudo-dynamic setup) during compression runs at different pressures. A high value of translational mobility coefficient[77] marks a region of cubatic phase. Mapping out the region of cubatic
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behavior was not possible for the 50% global-composition system as the cuberich phase formed by phase separation at even the lowest pressure already had the cubes in a cubic lattice (with φ ≈ 0.59). We therefore performed additional simulations at 60% to 90% number composition of cubes at pressures just below the phase separation pressure. This helped us better define the isotropic-cubic and isotropic-cubatic boundaries in the phase diagram.
The approximate Pressure vs. Composition phase diagram is shown in Figure 3.4. Note that even the ordered phases of each of the individual species allow a certain extent of mixing with the other species. Further, the cube-rich phase solvates significantly more spheres than the sphere-rich FCC phase solvates cubes. This asymmetric solvation capacity seems to be primarily the result of both our choice of particle size ratios which gives cubes a larger volume (and excluded volume) than that of spheres, and the geometry of the respective lattice spacings. Indeed, a sphere can readily replace a cube without overlap (if placed at the same center of mass) but not the other way round. Consequently, mixing entropy favors cubic phases where numerous spheres are allowed provided they do not take away the packing entropy gains associated with the overall cubic order. Conversely, cubes are only allowed in the FCC sphere-rich phase as long as they can appear as very dilute localized defects that do not compromise the overall structural order.
In the bigger picture, this mixture can be seen to represent a ‘base case’, of particles with very distinct shapes and incompatible lattice structures. In this scenario, although there is certain amount of mixing at intermediate concentrations, the ordered phases and the transition pressures remain quite similar to those of the pure components. This is expected since the two shapes are dif-
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ferent enough that the mixing entropy to be gained from their intermingling is outweighed by the packing entropy lost due to the incompatibility of lattices.
3.4.2 Cubes + Truncated Octahedra (CTO) Mixture
This mixture represents a case of two space-filling polyhedra, which have incompatible crystal structures. While cubes form a cubic lattice in their ordered state, truncated octahedra (TOs) form a BCC tessellation. Further, pure TOs exhibit a rotator mesophase as an intermediate between the liquid and the ordered crystal phase. Figure 5 shows the main results for this system with Fig. 3 also showing some relevant scatter plots (in the top row).
Upon extensive compression and expansion MC runs, the CTO mixture is observed to form a uniform isotropic phase at low volume fractions up to P∗ ∼ 7.7 where two phases emerge: a cube-rich cubic crystal and a TO-rich isotropic phase.
At P* ≈ 18, the TO-rich phase is seen to transition into a rotator mesophase, a value larger than that for pure TOs which form a rotator mesophase at P* ≈ 14. This difference is expected since the presence of cubes in the TO-rich phase should make ordering more difficult and drive the ordering pressure upwards. The onset of the positional order in the rotator mesophase is pinpointed by calculation of Q6 order parameter, which shows a sudden jump in Q6 for the TOrich phase.
At P* ≈ 50, the TO-rich phase achieves a crystalline BCC order (found through Q4 and Q6 values) and orientational order (found through P4 values
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Figure 3.5: Summary of results for the CTO mixture. a) Equation of state with ODPs for the individual particles marked for reference. Here ODPA is the ODP of TOs while ODPB is the ODP of cubes. b) Plot of P4 and Q6 order parameters. c) Pressure vs. composition phase diagram with different regions marked with the order of the constituent phases (xB is the mole fraction of cubes). d) Snapshots from the interfacial simulations of three representative regimes of the phase diagram (TOs are purple and cubes are dark red). Bottom: cubic+isotropic phases at P* ≈ 9. Middle: cubic +rotator phases at P* ≈ 25. Top: Cubic + BCC phases at P* ≈ 55.
and orientational scatterplots), a pressure that is again larger than that for pure TOs which gain crystalline BCC at around P* ≈ 28. To better define the cubatic region, we again perform independent runs away from the 50% number composition (between 80% and 95%) and assigned a cubatic character to systems exhibiting both high P4 values and a high particle translational mobility.
As seen in Figure 3.5(c), neither the cube-rich nor the TO-rich crystal phases allow a significant concentration of the other species. The saturation composi-
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tions tend to be symmetric; e.g., at intermediate pressures the cube-rich phase saturates with approximately 1% TOs and the TO-rich phase saturates with about 1% cubes. In a way, the CTO mixture shows less inter-species miscibility than the CS mixture. This could be because: (1) a slightly larger size disparity reduces the entropic gain obtained from mixing entropy as the pressure increases, and (2) the space-filling TOs are less tolerant of impurities (cubes) than spheres (which are non-space filling). This question is revisited in Section 3.5.
3.4.3 Cuboctahedra + Cubes (COC) Mixture
The components of this mixture have some key similarities and differences with those of the CTO mixture. Like TOs in the CTO mixture, cuboctahedra (COs) in the COC mixture also exhibit a rotator mesophase over a significant range of volume fractions, before going into a crystalline phase. The crystalline phase for pure COs, however, is a distorted cubic phase unlike the BCC crystal exhibited by pure TOs. More importantly, the two species in the COC mixture are quite close to each other in size, with a volume ratio of 0.83 (in the CTO mixture this ratio is 0.5). Our main results for this system are shown in Fig. 6 (see also Fig. 3 for some sample scatterplots).
Compression and expansion MC runs show phase separation of the mixture at P* ≈ 9, just above the MCP for cubes. Hence, phase separation upon compression of the isotropic phase gives rise to a CO-rich isotropic phase and a cube-rich cubic crystal. At P* ≈ 14, the CO-rich phase transitions into a rotator mesophase, as evidenced by a sudden rise in the value of the Q6 order parameter. For pure COs, this transition happens around P* ≈ 7. As the pressure increases to around
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P* ≈ 20, the CO-rich phase undergoes another phase transition from rotator to crystalline phase, while the same transition happens at P* ≈ 14 for pure COs.
An important difference observed in the Pressure vs. Composition phase diagram for the COC mixture (Fig. 3.6 (c)) as compared to the earlier mixtures is the relatively large saturation concentration of COs in the cube-rich phase (reaching about 30%) even at high pressures. This enhanced mixing may partially arises due to the COs gaining packing entropy by following orientations compatible with those of neighboring cubes. Further, near-equal volumes of the two phases imply that the packing entropy cost of allowing a CO-impurity in cubes is minimal. As in the previous systems, the low solubility of cubes in CO-rich phase is related to the bigger size of the cubes that makes them hard to be accommodated as guests in the CO-rich phase at high densities. See further analysis in Section 3.5.
3.4.4 Cuboctahedra + Truncated Octahedra (COTO) mixture
The COTO mixture represents a case where the two shapes involved are similar in terms of their individual phase behavior. Further, they are neighbors in shape evolution (during polyol process). While truncated octahedra form a BCC tessellation, cuboctahedra arrange in a distorted cubic lattice which is nonspace-filling. However, both shapes show a rotator mesophase for a wide range of volume fractions.
While one could have expected these two shapes to be the most miscible of the mixtures considered here, our simulation results shown in Fig. 7 (and in the relevant scatterplots of Fig. 3) provide evidence to the contrary. This is primarily
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Figure 3.6: Summary of results for the COC mixture. a) Equation of state with ODPs for the individual particles shown for reference. Here ODPA is the ODP of COs while ODPB is the ODP of cubes. b) Plot of P4 and Q6 order parameters. c) Pressure vs. composition phase diagram with different regions marked with the order of the constituent phases (xB is mole fraction of cubes). d) Simulation snapshots from three representative regimes of the phase diagram (COs are green and cubes are dark red) . Bottom: isotropic mixture phase at P* ≈ 6. Middle: cubic + isotropic phases at P* ≈ 10. Top: Cubic + distorted cubic phases at P* ≈ 25.
because although the similarity between shape and individual phase behavior could promotes miscibility, the size ratios adopted here (based on the polyol process) make cuboctahedra significantly bigger than truncated octahedra. Both the volume ratio (1.6) and the ratio of the radii of their circumspheres (≈ 1.25), translate into a twofold difference in the components? ODP values (see Table A.1) and a scant miscibility.
MC compression runs, which start from an isotropic liquid state fail to
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phase-separate and instead get kinetically trapped into a disordered state. It appears that at pressures where the mixture would prefer phase-separating, the mixtures is already too dense for a well-mixed system to demix and phaseseparate. However, expansion runs started from two-phase ordered configuration (with individual shapes ordered in their corresponding crystal lattices) is observed to be stable in a two-phase state down to very low volume fractions. We thus use expansion runs in this case to estimate thermodynamic phase behavior. The fact that the compression and expansion runs yield different results at high pressures signals lack of ergodicity (assumed to be more severe for the compression process). While unphysical MC moves like particle-swaps help overcome diffusion barriers to compositional equilibration, more elaborated moves may be needed to speed up structural equilibration (and the nucleation of translational order).
Below P* ≈ 30 (which is slightly above the MCP for TOs), the TO-rich phase goes into a rotator phase while the CO-rich phase is still ordered in a distorted cubic lattice. Below P* ≈ 18, again slightly above the MCP for COs, the CO-rich phase too becomes a rotator phase. In a narrow region between P* = 13.5 and P* = 11.5, we see the TO-rich phase in an isotropic phase while the CO-rich phase remains as a rotator mesophase. Below P* = 11.5, the two phases mix to form a single isotropic phase.
As far as inter-species miscibility, in the small region (between P* of 11.5 and 13.5) where some miscibility occurs, COs are seen to be more miscible in the TO-rich phase than the other way around. This is expected since in that region the TO-rich phase is largely isotropic which makes the entropic cost of introducing a CO impurity in the TO-rich phase minimal. On the other hand, the
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Figure 3.7: Summary of results for the Cuboctahedra-Truncated octahedra mixture. a) Equation of state showing the ODPs of the individual particles for reference. Here ODPA is the ODP of TOs while ODPB is the ODP of COs. b) Plot of P4 and Q6 order parameters. c) Pressure vs. composition phase diagram with different regions marked with the order of the constituent phases (xB is the mole fraction of COs). d) Simulation snapshots of three representative regimes of the phase diagram (COs are green and TOs are dark purple) . Bottom: rotator + isotropic mixture phase at P* ≈ 12. Middle: distorted cubic + rotator mixture phase at P* ≈ 20. Top: distorted cubic + BCC phase at P* ≈ 32.
CO-rich phase is positionally ordered and loses some packing entropy by hosting TO-impurities. This mixture demonstrates the fact that along with shape and individual phase behavior, the relative size of the particles can have a dominant effect on the mixture phase behavior.
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3.5 Discussion of general trends and comparison to other sys-
tems
Systems CS, CTO, and COC constitute similar mixtures of a large component B, cubes, with a smaller component A that has moderate asphericity (γ <1.5) and high rotational symmetry. The COTO mixture is slightly different but can also be cast as a A+B mixture with the TO and CO corresponding to species A and B. Overall these mixtures seem to exhibit a eutectic type of phase behavior, akin to that formed by binary A+B mixtures of hard spheres whose size ratio α (=diameter of A/diameter of B) is different enough (say α ≈0.85) so that at high pressures they tend to phase separate into two incompatible FCC lattices[137][138]. Figure 3.8 shows a qualitative diagram for such a case along with one that encapsulates the behavior observed in the CS, CTO, and COC systems. Note that our simulations were unable to resolve neither the position of the eutectic point (which in all cases seems to be near zero cube composition) nor the very small isotropic+solid A phase coexistence region. Despite the quantitative disparities in the diagram proportions and the differences in the lattice symmetry of the ordered phases, the physics underlying the phase diagrams (a) and (b) in Figure 3.8 is the same: just like in our systems, in the mixture of hard spheres phase separation at high pressure is driven by the maximization of packing entropy that results from forming two distinct efficiently-packing solid phases. At intermediate pressures where (only) the pure B system would solidify (as it has the lower ODP), a B-rich ordered phase must form which hence coexists with an isotropic phase within a large two-phase region. In this context, the fact that in our systems one or the two components have flat facets does not fundamentally change the overall picture, that merely leads to ordered structures that depart from the
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Figure 3.8: Qualitative sketches of pressure-composition phase diagrams for a binary mixture of hard spheres of diameter ratio α=0.85 (a), and for the CS, CTO, COC, and COTO mixtures (b). I = Isotropic, S A, S B = rich A and rich-B solids respectively, with A(B) being the smaller (larger) component
ones favored by spheres. Also, the fact that our systems exhibit mesophases (that precede the perfect crystal at high pressures) does not add anything fundamentally new in the character of these diagrams since all mesophase-crystal phase transitions are continuous and hence such mesophases simply occupy the lower-pressure portion of the A-rich or B-rich solid regions.
Another general trend relevant to the CS, CTO, and COC mixtures (excluding now the COTO mixture) relates to the correlation between component relative sizes and inter-phase solubility (i.e., how much of A can be dissolved in a B-rich phase and vice versa). Here cubes (the common component) can be taken as the reference whose size sets not only a reference unit length but also a common baseline to directly compare pressure values. In such a case, the pure components rank in order of decreasing size ( ∼ volumes) as cubes, COs, spheres, and TOs, which is also the order of increasing ODPs. This shows that the ODPs are more strongly affected by differences in particle size (than in particle shape).
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Figure 3.9: Pressure-composition phase diagrams for the CS, CTO and COC mixtures overlaid on top of each other. For simplicity, only the main two-phase envelopes are shown, leaving the boundaries connecting to the pure-species ODPs untraced . Two particular P* values, P* = 10 and P* = 18 are marked for reference. xB is the mole fraction of cubes.
To examine how the saturation solubilities compare for the same pressure across mixtures, we ovelay the pressure-composition diagram for these 3 mixtures in the same plot (Figure 3.9) and consider (for concreteness) two cases: P*=10 for the isotropic-cubic coexistence region and P*=18 for the rotator-cubic coexistence region (selecting other pressures would give similar results).
For a fixed P* we will denote as the most compressed A-rich phase (among the 3 mixtures) the one that has the lowest ODPA. (i.e., the degree of compression is relative to ODPA). This is rooted on the fact that at the ODP, the isotropic phase for all systems considered has a similar packing fraction and so one can assume that at the ODP all A-rich (or B-rich) systems are comparably dense. We see in Figure 9 that at a given pressure: (1) Cubes tend to dissolve more into
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the A-rich phase (whether it is isotropic or rotator) for the mixture that has the highest ODPA, namely, where the A-rich phase is less compressed (at the given P*), and (2) conversely, the A component dissolves more into the cubic phase for the mixture that has the lowest ODPA, namely, for the A-rich phase that is more compressed (at the given P*). Both of these trends are consistent with the idea that a component that is in a more compressed phase has a higher chemical potential (and a higher tendency to escape) and would then have a higher relative proclivity to transfer to the other phase so that chemical potentials can be equalized. For trend (1) cubes in the cubic phase are always equally compressed (for a given P*) but will transfer to and populate more a coexistence phase that is less compressed and hence more hospitable to guests; for trend (2) the A component experiences different states of compression in the A-rich phases across mixtures and hence escapes to different extents into the coexistence cubic phase (that is always equally compressed at a given P*). Because our systems are purely entropic, a higher (lower) relative degree of compression or chemical potential translates to essentially a lower (higher) entropy. Of course, these trends are only approximate and expected to hold provided that, among different systems, the ODPs are sufficiently different but one is still comparing at conditions where similar phases coexist. Note that other pure-system attributes (besides ODPs) could be used to try to correlate the observed miscibility trends; e.g.,the particle volume which, as Table 1 shows, consistently decreases as ODP increases and hence it is an equally good descriptor. Further, some particular function of ODP or particle volume may prove to be better at correlating those trends in a more quantitative way. Our limited sample of mixtures only allows us to point out qualitative correlations and to conjecture that relative values of ODPs, landmarks of pure component phase behavior that also capture particle
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size disparities, likely provide more robust clues of mixture phase behavior than any single geometrical feature of a particle shape. Of course, the precise particle shapes must also play a role in reinforcing or opposing these broad trends. The COTO mixture does not allow a direct comparison with the other mixtures for a fixed pressure but we expect that similar principles should apply on how the relative compression states of the phases affect the saturation solubility of the guest component.
3.6 Conclusions and a roadmap for equimolar phase behavior
Towards the goal of identifying key factors that govern the self-assembly of mixtures of polyhedral nanoparticles, we studied here the phase diagrams of four representative binary mixtures of hard convex polyhedra, using as components particle shapes that are readily accessible via well-established synthesis methods. In particular, we examined how the properties of the individual components affect the interplay between mixing and packing entropy which ultimately determines the types of phases formed and the extent of inter-particle mixing in such phases.
We find that, while the pure-component phase behavior is determined by the rotational symmetry and asphericity of the particle shape, the binarycomponent phase behavior depends on both the pure-component phase behavior and on the relative size ratio of the components, which in turn determines their relative difference in ODPs. For instance, in the COC mixture a combination of a similarity of size ratio and pure-component crystal lattices makes COs particularly more miscible in the cube-rich ordered phase.
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Casting our systems as A+B mixtures where B is the component with the largest size and hence smaller ODP, we expectedly find that the relative extent of miscibility of the two shapes (miscibility of A in the B-rich phase vs. that of B in the A-rich phase) depends on the relative ODP values. In particular, if ∆ ODP = ODPA − ODPB with ODPA > ODPB, then at any particular pressure where phase separation occurs, the larger the ∆ ODP the lower the solubility of A in the B-rich ordered phase and the higher the solubility of B in the A-rich ordered phase
We attempt now to sketch out a rough phase roadmap that identifies the phases formed at the ODP of an equimolar mixture of hard particles (henceforth denoted the ODPEM); i.e., the first single- or two-phase state involving at least one ordered phase that arises upon compression of an isotropic equimolar mixture of A+B. We only consider mixtures consisting of shapes that for any particular asphericity γ exhibit (as pure components) one or more of the following mesophases or ordered states: rotator (R), solid crystal (S), and liquid crystal (LC). If a given particle forms multiple ordered states, it is assumed that a LC occurs at a lower packing fraction (and pressure) than an R phase, which in turn would occur at a lower packing fraction than an S phase. Very high γ values are assumed to be accessible with prolate or oblate shapes only, which should lead to LC behavior and low ODPs. The tentative roadmap shown in Figure 3.10 is based on observations from this work and those from selected previous studies on binary mixtures of hard particles (marked by numbers in the plot), and is restricted to intermediate particle volume ratios r between 0.5 <r< 0.85, which is the range that our mixtures fall into.
The diagram of Figure 3.10 is guided by the following observations. If B is
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Figure 3.10: Tentative sketch of road map of phases at the ODPEM (the ODP of an equimolar binary mixture) of A+B hard particles for components with volume ratio 0.5 <r< 0.85. I = isotropic, LC= liquid crystal, R = rotator solid, S = crystal solid; a subscript denotes the majority component in the phase. Gray region in (a) corresponds to single compound solids. Numbered circles denote selected states studied in the literature (cases 2 through 5 are from this work) as follows: 1 = Ref. [137], 6 = Ref. [129], 7 = Ref. [139] and [130], 8 = Ref. [140], and 9= Ref. [141]
the largest component, then it is expected that for low to moderate γA, ODPA > ODPB and hence at the ODPEM (slightly above ODPB) a phase separated state should ensue comprising an ordered B-rich phase (R, S, or LC depending on the γB value) and an isotropic phase; however, for very large γA the pure A component would be expected to form a LC with ODPA < ODPB in which case at the ODPEM an A-rich LC phase should coexist with a B-rich isotropic phase.
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Of course, a crossover behavior could exist between these small-γA and large-γA regimes, where two ordered phases coexist at the ODPEM. In Figure 3.10 we mark only the ordered phases that could be formed, if any such phase will form at all. The alternate outcome would be the formation of some type of jammed state without a well-defined structural order. The shape and extent of each region are only qualitative and meant to guide the eye. A secondary particle shape parameter besides γ would be necessary to make more discriminative diagrams ( rotational symmetry would be a good candidate [77]). Note that we assumed that as γ approaches 1, particles have higher rotational symmetry and we ascribed to crystal phases of spheres a rotator character since any infinitesimal departure from γ = 1 would lead to a rotational degree of freedom.
Compound crystal phases are known to exist for a number of binary hardcore particles, like the Laves phases for unequal hard spheres[142] [106]or polyhedra that form tessellating compounds[129]. However, these may not be the phases that arise at the ODPEM (i.e., at equimolar composition) or may arise for components with r < 0.5 and so they would not be included in Figure 3.10. Likewise, a single mixed LC phase would only be a possibility for very specific types of component shapes. These two scenarios (where a single S or LC phase forms at the ODPEM) do not seem to correlate strongly with sphericity (other than a loose tendency of components to have similar γ) and so are only included in Figure 3.10 (as a gray region) for completeness.
Altogether, our observations highlight the fact that although many factors such as relative size, asphericity, individual crystal lattices, and mesophase formation determine the phase behavior of a mixture, the trends in mutual miscibility are best captured by the components asphericities and their relative ODPs.
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Towards designing novel nanoparticle superstructures with desired properties, this study hence provides some guiding principles about the phase behavior of the binary mixtures derived from the properties of the constituent shapes. While all systems studied here are relatively asymmetric in terms of size (and ODP) values, we are currently exploring more symmetric binary systems where our preliminary results have already revealed significant differences with some of the trends observed here. Also, while the systems studied here involved convex particles only, the use of concave particles, especially when paired with complementary-shaped convex partners, would open the door to much more complex phase behaviors.
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CHAPTER 4 ODP-TUNING AND MIXED ROTATOR MESOPHASES
4.1 Introduction
Polyhedral1 colloidal nanoparticles are versatile building blocks towards designing novel materials with targeted emergent properties. Recent developments in experimental techniques[38, 39, 63, 64, 68, 100, 144] to controllably synthesize and manipulate polyhedral nanoparticles have fueled many theoretical [84, 85] and simulation studies[77, 80–82, 87–92, 145] to understand their packing and phase behavior. These building blocks have been shown to exhibit a rich phase behavior at finite osmotic pressures unveiling the presence of novel mesophases. A mesophase is a partially ordered phase whose properties are intermediate between those of disordered liquids and ordered crystals, such as liquid-crystals, plastic-crystals, and quasicrystals.
Binary mixtures of polyhedra[124] exhibit a competition between mixing and packing entropy that often leads to phase separation at high pressures. Although ordered superlattices are desirable as a platform to create a wide array of composite materials, assembly into binary superlattices using just entropic forces is difficult to achieve [129]. An earlier study[124] on the miscibility trends of binary polyhedra mixtures revealed the importance of the relative size ratio of the components and of similarity in their mesophase behavior[77]. A plastic crystalline rotator phase is ubiquitous for shapes with small anisotropy and high rotational symmetry[77]. One of the aims of this work is to identify conditions
1This chapter is reprinted from reference [143] as follows: Mihir R. Khadilkar and Fernando A. Escobedo, Physical Review Letters, 113, 165504 (2014). Copyright (2014) with permission from the American Physical Society.
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that favor the formation of rotator mixtures for appropriately chosen component shapes and sizes, even in the absence of any aiding enthalpic interaction.
A family of truncated cubes, which is readily synthesizable [38,39], has been recently shown to exhibit a diverse set of phases[92]. Further, the kinetics of the disorder-to-order transition for some members of this family has been shown to be substantially faster than that of hard spheres[146], making them appealing choices for applications requiring fast self-assembly. In addition to cuboctahedra (COs) and truncated octahedra (TOs), we choose here a truncated cube with truncation parameter 0.4 (TC4) (a cube with 80% of its corners cut-off; see [92] for details), since, like COs and TOs, TC4 also exhibits a rotator mesophase[92]. Our focus on components with rotator mesophases is motivated by the hypothesis that mesophasic partial disorder can provide enough structural leeway to facilitate ordered solutions to form despite the entropy costs associated with differences in packing. The main mixtures studied are the three possible pairings of these three shapes, and are denoted henceforth as COTO, TC4TO and TC4CO.
4.2 Role of relative size-ratios and mesophases
For two components whose pure solids have different lattice motifs, the relative size-ratio is an important determinant to control the crystal lattice spacing. A key question is to find a size ratio that optimizes mixing in the rotator phase. A recent study[124] suggested that the miscibility in a binary mixture of polyhedra can be effectively linked to the relative values of the order-disorder transition pressure or ODP. Given that the osmotic pressure controls the concentration of the suspensions, the difference in ODP between the two pure-components, ∆
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ODP, would capture the difference in their proclivity towards ordering from the disordered state. In that study [124], however, the components’ ODPs were always substantially different and very limited solid miscibility was observed; hence, the questions of what happens when the ODPs matched and whether that provides optimized mesophase compatibility were left completely open. Hence, in our simulations, we set the relative particle size ratios such that their ODPs are approximately equal, which coincidentally entail near-equal circumradii; namely the ratios of circumradius are CO:TO = 1:1 for COTO, TC4:TO = 1.01:1 for TC4TO and TC4:CO = 1.01:1 for TC4CO (since not all vertices of TC4 lie on a single spherical surface, we use the largest circumscribing radius). While equal circumradius is an equivalent criterion to ∆ODP=0 for the main mixtures considered here, we will also use a fourth mixture of spheres and cubes to show that ∆ODP=0 optimizes the overall miscibility even for solid phases other than the rotator mesophase.
For the main mixtures, we probed the equilibrium phase behavior as a function of pressure using hard-particle Monte Carlo simulations in the isothermalisobaric ensemble, using moves for thermal, mechanical, and compositional equilibration, including swaps between the position of particles of different species[124]. We used interfacial runs to test the relative stability of the phases near a phase transition. While most simulations used equimolar mixtures, additional runs for other compositions were used to more completely map out the phase diagram. Orientational order was analyzed by using the P4 order parameter [86] and orientational scatterplots (see appendix A), while the translational order was probed by using Steinhardt’s order parameters Q4 and Q6[113] and diffraction patterns (structure factors). This information was used to detect phase boundaries. To further characterize positional order, we also identified
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the contributions of FCC, BCC or HCP-like motifs [77] by calculating the distributions of two local bond order parameters (q¯4 and q¯6), and comparing them to those for the reference liquid, HCP, FCC and BCC structure phases (see details in the appendix A).
4.3 Results
The COTO, TC4TO, and TC4CO mixtures exhibit a rotator mesophase in between the isotropic phase at low pressures and a phase separated state with two crystalline phases at high pressures (see Figure 4.1). This mixed rotator mesophase (MRM) is stable for all compositions in all three mixtures and for a sizable range of volume fractions (see appendix A). It is of interest to characterize such novel MRM since the rotator phases of the pure components are distinct in both translational order and rotational disorder. For instance, after the ODP the COs and TC4s rotator phases transform into the orientationally ordered crystal via a first-order transition at the mesophase-to-crystal transition pressure [92]; in contrast, TOs transform continuously into a crystal phase. Indeed, while P4 remains low and relatively constant for the pure CO and TC4 rotator phases as pressure increases, it continuously increases for the pure TO rotator phase[146]. Below we examine the properties of this MRM in more detail, giving representative results for the COTO mixture.
In a purely entropic scenario, mixtures (that do not form tessellating compounds[129]) would be expected to phase separate at high pressures into nearly pure component solids to allow denser packings. For our ODP-matched mixtures, the packing incompatibility between shapes is minimized and the onset
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Figure 4.1: Pressure (P*) vs. composition (xB) phase diagram for the 3 main mixtures. DSCC and DSCT are distorted simple cubic structures of COs and TC4s respectively[81, 92]. The number fraction xB represents fraction of COs in the COTO and TC4CO mixtures, and the fraction of TOs in the TC4TO mixture. Each phase diagram is accompanied by a snapshot of the mixed rotator mesophase (MRM) for xB=0.5 (and P*=11.2, 9.6 and 9.6 for COTO, TC4TO and TC4CO respectively), its orientational correlation plot and the diffraction pattern.
76

of phase separation is hence delayed (e.g., P* ≈ 21 in the equimolar COTO). The observed MRM has intermediate orientational order ( P4) as shown in Figure 4.2-a for the COTO mixture, and strong positional order (Q4 and Q6). Local compositional heterogeneity or incipient ‘clustering’ in this MRM can be detected by calculating the fraction of like-shaped nearest neighbors to a given particle species. This fraction should equal the overall composition of the given species in the bulk system for an ideal mixture, but it will be larger than that as clustering and a tendency for phase separation ensues. We observe that for all three mixtures the ratio of local to global composition or ‘enrichment factor’ ( f ) steadily increases with pressure from its ideal (well-mixed) value until eventually reaching the solid-solid phase separated state (Figure 4.2-b). The more symmetric mixtures have larger ideal mixing entropy and hence enrichment factors closer to unity. For some of the more skewed compositions (away from 50%), the MRM crystallizes before phase-separating as pressure increases. Figure 4.2-a shows how the mesophase-to-crystal transition (as detected by the approach of P4 to the threshold value of 0.4 for orientional order) changes from being nearly continuous for low CO-compositions (similar to pure TOs) to having more abrupt increases for higher CO-compositions (like pure COs).
Given that none of the MRMs simulated had one of the known perfect lattice structures, we obtained the fractions of different standard structural motifs in the simulated configurations [77]. We observe that in the equimolar MRMs containing TOs (COTO and TC4TO), the fraction of BCC (which is the target structure for TOs, the better-packing shape in the mixture) increases with volume fraction (see appendix A). Similarly, for TC4CO, the MRM at lower volume fractions is composed of BCC and HCP, while the fraction of HCP (which is closer to DSCC and DSCT structures that COs and TC4s favor respectively) increases with vol-
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Figure 4.2: Plots showing the effect of changing the mesophase composition in COTO mixture (solid part of each curve represents the stable MRM region). (a) Variation of P4 (averaged over all particles) as a function of pressure, P* . (b) Pressure dependence of the enrichment factor (change in the neighbor fraction of COs normalized by its ideal well-mixed value).

ume fraction.

Table 4.1: Summary of results for the miscibility range for COTO mixture in the original and changed size ratios Vl and Vs correspond to the volume of the larger and smaller particle in the mixture respectively. El and Es denote edge lengths while Rl and Rs denote circumradii.

System ∆ ODP ∆P∗m ∆φm AMRM Vl/Vs El/Es Rl/Rs

O

≈ 0.0

13 0.17 7.0

1.21 1.58 1.0

S

≈ +1.1 9.0

0.13 6.1

1.04 1.66 1.05

L

≈ -1.0 3.0 0.09 3.6

1.41 1.50 1.05

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4.4 Importance of ODP-tuning
To test whether the equal ODP rule maximizes solid-state miscibility, we use the COTO mixture as testbed and change ±5% the relative size ratio by slightly perturbing the size of TOs from its original value (assumed unity, system O), to be 1.05 ( system L, for larger TOs) and 0.95 (system S, for smaller TOs). This rescaled the ODP of the corresponding TOs from 7.1 (system O) to 7.1 × 0.953 = 6.1 (system L) and 7.1 × 1.053 = 8.2 (system S). The first observation is that systems L and S also exhibit an MRM over the whole range of compositions, showing that this MRM behavior is robust to small changes of particle size ratios (e.g., size polydispersity that may arise from the experimental synthesis). The extent of miscibility in the MRM can be quantified by using several metrics, e.g.: (1) ∆P∗m: The difference between the highest and lowest pressure where the equimolar MRM phase is stable, (2) ∆φm: The difference between the highest and lowest volume fraction where the equimolar MRM phase is stable, and (3) AMRM: The area where the MRM exists in the volume fraction vs. composition phase diagram. We observed that the extent of miscibility as inferred from all metrics decreased for systems L and S relative to system O. Indeed, the MRM ∆P∗m stability range was ≈ 13 for system O, ≈ 9 for system S and ≈ 3 for system L. Further, in a previous study[124] where the size ratio was 63 % the ODP-matching value, no MRM formed for a wide range of compositions.
While the CO:TO volume ratio is not a good predictor of MRM miscibility as it is closer to unity in the L case than in the O case (See Table 4.1), the ratio of circumradii is. Equal-circumradii, which also holds for the TC4CO and TC4TO mixtures described earlier, could be envisioned as allowing two low-asphericity polyhedral components to freely rotate, effectively sweeping equal spherical
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volumes in the lattice sites of the MRM. This picture is too simplistic, however, since TOs do not freely rotate in their mesophase[146].
To discriminate the role on mixture phase behavior of particles with equal ODP vs. particles with equal circumradius, the components should not both be ‘round-shaped’ but one of them have high asphericity. For contrast, we simulated mixtures of spheres and cubes. Spheres can be seen as the limiting case of a rounded polyhedra, whose FCC solid can also be taken to be a rotator if a minimal shape anisotropy is assumed [147]. Cubes can be seen as the limiting member of the truncated cube family having minimal truncation and high asphericity, whose solid phase is no longer a rotator[77]. Figure 4.3 shows the phase diagrams traced using a Gibbs-Duhem integration method (see appendix A and ref.[148]). Results are shown for 3 choices of the sphere diameter σ to cube edge d ratios: 1 (equal inradius), 1.23 (equal ODPs), and 1.732 (equal circumradius). Equal circumradii leads to minimal mutual solid solubility and an almost non-existent MRM region. In contrast, equal ODPs lead to maximized mutual solid miscibility with both a large region where spheres dissolve in the cube-rich solid (C region) and a large MRM region where cubes dissolve in the sphere-rich solid (S region in gray) . In that latter MRM, the orientation scatterplot (Figure 4.3 ) reveals that cubes form a restricted rotator (Figure 4.3) where they lack orientational order but cannot adopt certain orientations. Such orientational correlations (e.g., see Figure 4.1) depend on the shape and size of the particle relative to those of the ‘cage’ where it rattles.[146]
The above analysis suggests that the ODP is a potentially more generally predictive parameter of solid-phase miscibility of two shapes (beyond rotator mesophases). The ODP can be seen as marking the turning point where packing
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Figure 4.3: Pressure-composition phase diagrams for spheres (diameter σ) and cubes (side edge d) with different size ratios. Top: σ/d=1.0 (equal inradius), center: σ/d=1.23 (equal ODPs), and bottom: σ/d=1.732 (equal circumradius). S = sphere rich solid, C = cube rich solid, I = isotropic phase; P∗ = Pd3/ . Data for σ/d=1.23 is from [148]. Orientation correlation plots are shown for the cubes in the 3 phases occurring at the eutectic pressure.
entropy takes over as the dominant entropic force determining the structure of the system. Accordingly, if the components have the same ODP, their tendencies to order will be comparable (i.e., synchronized) at any pressure above this ODP. For pure systems having a rotator mesophase, matching ODPs in a mixture effectively synchronizes their mesophases along the scale of the thermodynamic field driving the phase transitions (i.e., pressure). Indeed, for A+B mixtures, if
81

ODPA ODPB then for ODPA < P <ODPB particles B will have a strong preference for the isotropic state, while for P > ODPB where both favor ordered states, particles A would be much more compressed than those of B and prone to form a separate A-rich dense solid. If one considers the pure components and that µ∗ = ODP(Z − 1)/PdP is the residual chemical potential of the isotropic phase
0
in coexistence with the ordered phase (Z is the compressibility factor), then for hard-core systems whose isotropic branches of the equation of state ( i.e. Z(P)) are similar (see Figure 1 in appendix A), having equal ODPs translates into pure mesophases that at the same pressure also have comparable chemical potentials and (neglecting the typically small ∆PV terms) similar entropies. If rotational entropies are also comparable (as in rotator phases), equality of ODPs then approximately translates into pure mesophases of A and B where each particle experiences a similar packing entropy or free volume: a likely helpful condition for co-assembly.
As a final test of the equal-ODP rule, we simulated a ternary equimolar mixture of CO, TC4 and TOs at ODP-matching ratios, and found that the ternary MRM phase is also stable (with ∆P∗m ≈ 3.6; see appendix A). Of course, equality of ODPs is not sufficient to ensure high solid-phase compatibility; similarity in the type of ordered structure is also important as with the rotator mesophase in the COs, TC4s, and TOs; in this context, the sphere-cube system provides a counter example where solid miscibility over all compositions is precluded by the different pure-component solid behavior.
Recent work from Van Anders et. al. [149, 150] described the assembly of anisotropic particles as driven by an entropic bonding arising from ‘entropic patches’ that is quantifiable via a potential of mean force and torque
82

(PMFT)(akin to enthalpic interactions). As the MRM is compressed and the patches get closer, any PMFT difference between like and dislike particles become more accentuated, making the mixed state less entropically favorable. This effect is connected with the changes in local composition that were discussed before in reference to Figure 4.2-b: like-particle contacts are favored with increasing density as though an effective attraction (repulsion) acts between the like (unlike) particle types. Eventually, the entropic cost at higher densities overpowers the mixing entropy leading to phase separation into two ordered phases. (This analysis does not apply to polyhedral mixtures that form tessellating crystals [129]).

4.5 Conclusions

Beyond polyhedral particles, binary mixtures of rigid rods (of diameters D1 and

D2 and lengths L1 and L2) with ODPs associated with isotropic-nematic tran-

sitions provide further insights. Simulation [151] and Onsager’s theory [152]

have shown that rods sufficiently dissimilar in length and/or diameter phase

separate into two nematic phases at high pressures (a sign of incompatibil-

ity).

However, ‘symmetric’ mixtures [153] where L2/L1

=

(D2

/D1)−

1 2

so

that

pure components have the same excluded volume and hence identical phase

behavior and ODPs, tend to lie well inside the predicted one-nematic phase

domain (see Figure 3 of Ref. [152]), with equimolar mixtures having compo-

nents with the same extent of orientational order (a sign of maximal compatibil-

ity) and ordering at pressures below the pure-component ODPs[153]. Further,

novel biaxial nematic phases have also been predicted for equal-ODP (symmet-

ric) blends of rod-like and plate-like ellipsoids [154–156]. Note that in these

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examples and our simulated systems, ODP equality is not a prescription that guarantees full mesophase miscibility (which could only happen when particle shapes and pure-component behaviors are not too disparate); instead, it provides a guideline for conditions that favor miscibility (even if only a partial one, as for the cube-sphere example discussed in Fig. 3).
In summary, we find that by choosing size ratios that synchronize the onset of the plastic crystals in the pure components of a mixture, fully mixed mesophases are favored despite incompatibilities in the lattice structure of the pure component crystals. A vast array of applications including optical devices[4], solar cells[5, 157–159] , photonic band gap materials, and metamaterials[160] will benefit from new routes to create nanoparticle superstructures. Just like liquid-crystal phases have found widespread applications as switches and sensors, it is plausible that rotator phases may also find applications involving the external control of their rotational state. Since components can be chosen to have different chemistries, the ability to produce rotator phases of any composition is certain to add to this potential.
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CHAPTER 5 PHASE BEHAVIOR OF POLYHEDRAL NANOPARTICLES IN PARALLEL
PLATE CONFINEMENT
5.1 Introduction
Anisotropic colloidal nanoparticles have been receiving significant attention in the scientific literature due to both their importance as model systems to study various atomic phenomena and their potential for technological applications. Much of this interest has been fueled by the development of robust synthesis methods[28, 30, 34, 36, 60, 127, 161] and the ability to tailor and control various key properties like shape, size, and the type and strength of specific interactions. In particular, regular polyhedral nanoparticles have emerged as versatile building blocks which can be efficiently synthesized[37–39] and give rise to a diverse range of phases[68, 77, 80, 82, 84, 86, 87, 89] due to their combination of shape anisotropy and rotational symmetry. Controlled assembly of these ordered structures is desirable in terms of technological applications, specifically towards tuning their structural, mechanical, electrical, photonic and dielectric properties.
Apart from the shape and interactions of the individual particles, phase behavior is also strongly affected by a strong spatial confinement that would preclude bulk behavior to fully develop. In fact, both experimental[162–168], and modeling[169–173] studies have found new particle structures that only form under confinement. Apart from its effect on the packing of simple spherical particles, spatial confinement has been shown[174] to control the ordering in various other shapes including hard rods[175], spherocylinders[176], poly-
85

gons[50, 177, 178] and spherical caps[179]. Spatial confinement poses an additional constraint on the assembly of particles, often giving rise to structural motifs not seen in the bulk. In the absence of any enthalpic interactions, when the ordering is driven only by entropic forces (which are constrained by the accessible space), the extent of spatial confinement molds the types of structures that maximize the total entropy. Even an exotic phase like a quasicrystal, which is otherwise not easily seen in bulk simulations, has recently been observed[180] using spatial confinement. Several established experimental techniques are routinely used wherein colloidal particles are confined to a few layers, thus giving rise to a range of phases depending on the extent of confinement[181–184]. Some of these confined systems have been shown to have interesting tunable optical properties including photonic band gap[183, 185, 186]. In this paper, we examine how spatial confinement affects the phase behavior of hard particles with selected polyhedral shapes from the truncated cube family[92]. A recent simulation study[94] explored the phase behavior of these shapes when pinned to a flat interface which mimics some experiments of particle assembly performed on a fluid-fluid interface. In that scenario, the centers of mass of the particles essentially exist on a 2D plane (even if particles can freely rotate). Our focus, however, is on slit confinement that allows for the formation of a few particle layers in between two hard walls; such a system allows bridging the gap between the known phase behaviors for such particles in two and in (bulk) three dimensions.
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5.2 Shapes, Model and methodology
Using Monte Carlo (MC) simulations, we studied four shapes from the truncated cube family [92] (see Figure 5.1) with increasing level of truncation, s (the precise definition of s is detailed in the Supplementary Information of Ref. [92]). Specifically, we studied perfect cubes (C) (s = 0), truncated cubes with truncation parameter 0.4 (TC4), cuboctahedra(CO) (s = 0.5) and truncated octahedra (TO) (s=0.66). Spatial confinement was modeled by hard walls in the Z-direction, while periodic boundary conditions were used in X and Y directions. Each shape was studied for a range of confinements, each of which was characterized by a non-dimensional height H* = H/ σ, where σ is the minimum distance between the confining walls that allows the particle to exist within the walls (in at least one orientation) without overlap with the walls. For C, TC4 and CO, σ is the edge-length of the equivalent cube that the shape is truncated from. For TO, it is the distance between the parallel hexagonal faces. Lengths reduced by σ are meaningful since they convey the maximum number of layers that a particular confinement value can potentially allow (barring buckled layers).
Figure 5.1: The shapes studied in this work, from left : (a) Perfect cube(C), (b) truncated cube with truncation parameter 0.4 (TC4), (c) cuboctahedra(CO) and (d)truncated octahedra(TO).
All shapes were treated as hard particles (with no enthalpic interactions), which amounts to disallowing any particle-particle or particle-wall overlaps (using the Separating Axis Theorem[132]). For each of the shapes, we simu-
87

lated a system of 800 particles for a range of H* values (from H*=1 to H*= 5). At high densities, our simulations spanned between 12 to 20 particle rows (depending on H*) along each of the X and Y dimensions to minimize finite-size effects. We also checked our results by performing simulations on larger systems (N=2000). For each of the H* value, MC simulations were carried out in a constant-pressure (NPT) ensemble, starting from low density liquid-like configuration allowing for slow compression via pressure steps. Each MC cycle consisted of N translation, N rotation and 2 volume move attempts. Each pressure step included 3 × 106 MC cycles for equilibration and 106 cycles for production. The volume moves affected the X-Y dimensions only and after the system ordered, were allowed to be triclinic (to relieve any remnant stress). In certain areas of the phase space where full equilibrium was suspect, we checked our results with longer MC runs (7 × 106 cycles per step).
5.2.1 Order parameters
We make use of several order parameters to track the formation of ordered phases in the self-assembly. Because of the geometry of our system, the usual order parameters used for either two or three dimensional bulk systems are not directly usable and do not carry the same importance or meaning in the current situation. Hence we use order parameters in a modified form as specified below.
To track positional ordering, we used the Ψ4 and Ψ6 order parameters that are generally used in two dimensions to track square or hexagonal bonding symmetry, respectively. To calculate Ψ4 and Ψ6, we assign for each particle i, a
88

complex number characterizing its local n-fold bond orientational order φn(ri):

φn(ri)

=

1 ni

ni
exp(inθi j)
j=1

for n = 4 and 6. Here, θi j is the angle made by the virtual bond between

particle i and its nearest neighbor j with respect to an arbitrary global axis, and

ni is the number of nearest neighbors of particle i. For n=6, ni is calculated via

Voronoi tessellation, while for n=4, the 4 closest neighbors are used for the cal-

culation to avoid the degeneracy in Voronoi construction that can arise in such

a case[187]. The global bond orientational order Ψn is calculated by:

Ψn =

1 N

N
φn(ri)
i=1

An important modification we make to the above definition of Ψn is that we isolate single layers within our simulations (in case there are more than one) and then calculate Ψn values for the set of particles on each layer, only counting in-layer neighbors. We define the particles to be within a particular layer by specifying a suitable Z-coordinate range, such that particles define a layer parallel to the walls and exclude any buckling.

Additionally, we also use the Q4 and Q6 bond-order orientational parameters [113] to probe and monitor translational order. These parameters are defined as:

1

Ql =

4π 2l +

1

 

+l

2 |Q¯ lm(r)|2

−l

where Q¯lm(r) is given by

Q¯ lm(r)

=

1 Nb

Ylm(r)
bond s

where Ylm(r) are spherical harmonics for the position vector r.

(5.1) (5.2)

89

Like Ψ4 and Ψ6, we modify the definitions for Q4 and Q6 as well, by restricting our summation to bulk-like particles (particles that are not in the layers closest to the walls).

Global orientational order is gauged by cubatic order parameter[86] P4, which is defined as:

P4

=

max n

1 N

P4(ui · n)

i

= max 1 n 8N

35cos4θi(n) − 30cos2θi(n) + 3

i

(5.3) (5.4)

where ui describes the unit vector describing the particle orientation and n is a

director unit vector that maximzes P4. However, for shapes with flat faces in

confined geometries, the director perpendicular to the walls tends to be associ-

ated with high P4 values even in the presense of in-plane disorder. Hence, we also use the director with second highest P4 value which more reliably tracks in-plane orientational ordering.

Different phases are identified by analyzing the trends and features of the relevant order parameters and the equation of state (EoS) as obtained from the compression runs. The phase diagrams for each of the shapes are obtained as a function of reduced pressure P* and volume fraction Φ. The reduced pressure P* is defined as P∗ = Pσ3/kBT where σ is the length used for non-dimensionalizing H*, kB is Boltzmann’s constant and temperature T=1.

5.3 Results
In our ensuing descriptions, we will focus on the ordered phases that form at high densitites (above the disorder-to-order transition pressure) and hence we
90

disregard the low-density isotropic fluid. Before describing results for specific particle shapes, we first outline some trends of behavior that are shared for all systems. In the absence of any enthalpic interaction, the assembly of the systems studied here is driven purely by entropy. Hence, the systems try to minimize their Gibbs free energy that, if not for a typically small PV contribution, essentially corresponds to maximizing the entropy for given conditions (pressure and confinement). For a confinement which is equal to or slightly bigger than a length commensurate with ∼ N layers of the expected particle lattice arrangement (i.e., H* ∼ N), the densest-packing assembly is nearly perfectly ordered and can attain very high density values. However, when H* is slightly less than a whole number, the system cannot pack space efficiently; this gives rise to relatively low densities at the densest-packing state where the unused space will allow some partial translational disorder. Further, such low densest packed states can produce structural motifs that would not be favored in the bulk systems. Overall then, H* controls the number of layers possible in the Z direction and the symmetry of the phases observed.
5.3.1 Cubes
We show the broad trends in phase behavior for cubes in a phase diagram that plots volume fraction Φ vs. degree of confinement H*(see figure 5.2). Broadly, we observe a transition from a disordered liquid at low pressures (and low volume fractions) to a crystal with square or cubic order as the system is compressed. This behavior is similar to those observed in both bulk 3D systems and 2D hard squares[177]. For the H* values explored here, we do not observe a cubic mesophase as seen in the 3D case[77, 86]; however, for confine-
91

ments approaching a single layer (H*→ 1), we see regions which have significant particle orientational order and intermediate four-fold bond-orientational order (gauged by Ψ4), a signature of tetratic order. The Ψ4 value for these confinements increases continuously with pressure, giving way to a square order (see Figures B.1 and B.2).
In the entropy-driven self-assembly of cubes, one observes cubic crystalline ordering at sufficiently high volume fractions. However, such perfect ordering requires commensurability of lattice spacing with the space available (in the simulation box). Under slit confinement, that requirement is not always satisfied along the Z axis. Hence the phase that maximizes the entropy at high volume fractions is not necessarily perfectly ordered cubic phase (since the extra space resulting from the incommensurability is used for additional configurations which leads to a partial disorder).
The dependence of phase behavior and multilayering on H* can be illustrated over different H* ranges (see Figure 5.3 for a range of sample snapshots). For example, up to H* = 1.9, there is space only for a single layer. So for confinements in the range 1.1 < H∗ < 1.9, the free volume available around the layer increases with H*, which leads to higher misalignment (in the Z-direction) as we increase H*. This can be explained by the fact that when the space available is not enough to accommodate another layer, the system maximizes entropy by exploring the configurational freedom available in the Z-direction (which leads to misalignment).
Within each regime corresponding to a particular number of layers, as we increase H*, there is more space available which leads to inefficient packing. These local misalignments represent local displacement modes akin to phonons
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Figure 5.2: Phase diagram of cubes as a function of volume fraction Φ and confinement H*. The boundaries marked here are only approximate, since it is difficult to exactly pinpoint coexistence regions in our simulations.
in a perfect crystal. It is also important to note that these modes persist at both intermediate and high densities. Because of the periodic boundary constraint, we cannot examine existence of modes of arbitrarily large periodicity.
As we go from H*= 1.9 to H* = 2.1 (thus increasing a layer), the packing becomes abruptly more efficient. With more than one layer, one can also envision displacement modes wherein one layer slides over another. But we did not observe any significant inter-layer displacement. This could be in part due to the fact that displacement modes in the Z-direction have to be coordinated within and across layers and inter-layer sliding (on the XY plane) would pro-
93

Figure 5.3: Snapshots depicting the structure of cubes at representative confinement values (a) H* = 1.5 at φ ≈ 0.61 (b) H*= 2.1 at φ ≈ 0.85 , (c) H* = 2.9 at φ ≈ 0.62 , (d) H* = 4.5 at φ ≈ 0.78 . Each of the snapshots depicts the extent of the ‘displacement modes’, and its dependence on H*
hibit this entropic freedom. The importance of these displacement modes, even in presence of more than one layers (in 2 < H∗ < 3) is evident when we look at progression of high-density configurations as we go from H* = 2.1 to H* = 2.9.
Overall, the phase behavior we observe for cubes in slit confinement for H*< 2 is broadly similar to those observed in earlier studies of squares in 2D[177] and of freely rotating cubes restricted to an flat interface[94]. However, crucial differences emerge from the fact that slit confinement results in more extra space available as layer spacing incommensurability increases, which results in par-
94

ticle in-plane misalignment and displacement modes akin to phonons in bulk crystal.
5.3.2 TC4
TC4 is a truncated cube with 80% of its corners cut off. We chose this shape since it has been shown to exhibit plastic crystalline behavior[92] in 3D bulk systems and is significantly different from either cube (s=0) or cuboctahedra (s=0.5).
We observe a phase behavior whose broad features are similar to those of cubes as Φ increases: disorder liquid giving way to distorted square crystal (see Figure 5.4). However, there are regions in the phase space where TC4 behaves differently. While we see a similar trends as those for cubes within a particular H* range associated with an integer number of layers, between H* = 1.9 and H* = 2.1 we see different phases altogether.
Near H*=1.9, we see a buckled rhombic phase (BR), wherein the particles get subdivided into either being close to the upper or the lower wall. They still do not have enough space to form a separate layer (nearest neighbor particles tend to be at different heights in Z-direction), and hence they form a buckled phase with rhombic bond-orientational order (see figure 5.5b).
In contrast, at H*= 2.0 when the confinement width is enough to allow two separate layers, we see the formation of a two-layer square rotator (2SR) phase (between Φ = 0.39 to 0.47), eventually giving way to two-layer square (2S) phase. The rotator phase is characterized by a reduced orientational order in spite of strong positional order. Above Φ = 0.47, the system attains moderate orienta-
95

Figure 5.4: Phase diagram of TC4 as a function of volume fraction Φ and confinement H*. The nomenclature for the phases is as follows: the number represents the number of layers; S or S D represent square and distorted square phases respectively while BR and S R represent buckled rhombic phase and square rotator phase, respectively.The boundaries marked here are only approximate, since it is difficult to exactly pinpoint coexistence regions in our simulations.
tional order, although some particles are still not oriented parallel to the wall (see Figure B.3 in appendix B).
As we go beyond H*= 2.0, the only ordered phase we see is a distorted square phase with all particles oriented parallel to wall ( any rotator phase is absent). Beyond H* = 2.0, the phase behavior follows an overall trend of decreasing densest packing until an extra layer can be added, which results in a jagged phase boundary similar to that seen for cubes. Although TC4s in the bulk exhibit
96

Figure 5.5: Snapshots depicting the structure of TC4s at representative confinement values: (a) Single layer distorted square (1SD ) phase at H* = 1.5 and φ ≈ 0.55, (b) Buckled rhombic (BR) phase at H*= 1.9 and φ ≈ 0.46, (c) Twolayer rotator square (2SR) phase at H* = 2.0 and φ ≈ 0.46, (d) Two-layer distorted square phase (2SD) at H* = 2.9 and φ ≈ 0.57. Particles are colored for ease of visualization and identification of layers only.
a rotator mesophase near the isotropic-solid transition, we did not observe a mesophase over the upper range of H* values studied here. This is likely due to having an insufficient number of layers in the Z-direction to approach bulk 3-D behavior.
97

5.3.3 Cuboctahedra (COs)
Going further in truncation parameter, we also examined cuboctahedra (s=0.5), which have all eight of their corners cut off. Like TC4s, cuboctahedra also show a rotator mesophase in their bulk 3D phase behavior[77, 92].
Figure 5.6: Phase diagram of CO as a function of volume fraction Φ and confinement H*. Green shaded areas represent two-phase coexistence regions.
Compared to TC4 or cubes, we observe a rather different and richer phase behavior for COs (see Figure 5.6). For small confinements ( 1 < H∗ < 1.5), we observe a distorted square phase just like TC4 and cubes.) However, for 1.5 ≤ H∗ < 1.7 we see a hexagonal rotator phase (1HR) leading to a buckled rhombic
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(BR ) phase similar to that seen for TC4s. A hexagonal rotator phase, that acts as a precursor to the BR phase, contains particles with a significant six-fold bondorientational order (Ψ6) while having little or no global orientational ordering (see Figure B.4 in appendix B). At high volume fractions, the particles in the BR phase exhibits some segregation along the Z-coordinate, without the formation of a separate layer (akin to the behavior of the BR phase in TC4s).
For 1.8 ≤ H∗ < 2.3, we again see a square rotator (2SR) phase, eventually leading to a 2-layer distorted square (2S D) phase. The 2SR phase is stable for a large range of volume fractions (e.g., for 0.38 < Φ < 0.58 when H* = 2). For 2.3 ≤ H∗ < 2.6, we do not observe a rotator mesophase, although we detect the formation of a distorted square phase similar to that observed for smaller H* values. In this region, the particles are in a frustrated state between 2 and 3 layers. Hence, we see many particles appearing as defects attempting to form a third layer.
For 2.6 ≤ H∗ < 3.0, we see a hexagonal rotator phase, 2HR (a rotator phase with six-fold bond orientational order but no global orientational order), that gives way to a distorted rhombic phase (2RD). In this case, while the first transition (isotropic to 2HR) is seen to be first-order, the second transition (2HR to 2RD) appears to be a continuous transition (see Figure B.5 in appendix B).
After H∗ ≥ 3, we see a regular trend in the phase behavior, between (3 < H∗ < 4) and (4 < H∗ < 5). Within each of these regions, for small values of H* the isotropic phase transitions into a square rotator phase S R, which has a high value of the Ψ4 order parameter, that upon further compression goes into a distorted square (S D) crystal. For larger values of H*, however, the isotropic phase transitions into a hexagonal rotator phase HR that upon compression gives way
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Figure 5.7: Snapshots depicting the structure of COs at representative confinement values: (a) Buckled rhombic (BR) phase at H* = 1.7 and φ ≈ 0.48, (b) Twolayer rotator square (2SR) phase at H* = 2.0 and φ ≈ 0.69, (c) Three-layer distorted square phase (3SD) at H* = 3.2 and φ ≈ 0.68, and (d) Three-layer distorted rhombic phase (3RD) at H* = 3.8 and φ ≈ 0.60. Particles are colored for ease of visualization and identification of layers only.
to a distorted rhombic crystal (RD). This change is likely due to differences in the availability of space in the Z-direction associated with H∗. Near the lower bound of H* (3 < H∗ < 3.4 and 4.0 < H∗ < 4.5) , the space is almost commensurate with a cubic-like ordering (leading to high four-fold bond orientational ordering). Near the upper bound of H∗ (3.4 <H∗ ≤ 3.9 and 4.5 < H∗ < 5.0), the extra space is not enough to accommodate an extra layer, but is large enough to significantly modify the preferred particle structural arrangements.
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Overall, we see a much larger variety of phases in COs compares to cubes or TC4s. While the buckled and square rotator phases are also observed for TC4s, only COs give rise to the distorted rhombic (RD) phase and hexagonal rotator (HR) phase. COs are also able to arrange into distinct structures in the regions where the is extra space in the Z-direction is relatively large but insufficient for an extra layer.
5.3.4 Truncated Octahedra (TOs)
All the particle shapes study so far are largely ‘cube-like’ and hence form cubic or distorted cubic lattices at high densities in the 3D bulk state[80]. TOs, , having the largest truncation parameter (s = 0.66) and the smallest asphericity (a = 1.29), form instead a space-filling BCC lattice in bulk 3D conditions.
A distinctive feature of a cubic or distorted cubic lattice is that it can be seen as the stacking of parallel layers of particles. This is important in a confined slitgeometry since the confining walls now introduce fixed dividing planes which can predetermine the orientation of the particle layers of a (near) cubic lattice. . For the BCC packing of TOs, however, any planar cut to the lattice leaves gaps in the lattice, thus leading to inefficient packing. This can be clearly seen from the maximum volume fractions (achieved in our MC compression simulations), that are significantly lower for TOs than those achievable for TC4 or cubes (e.g., for H∗= 2.2, φmax values for cubes, TC4s, COs and TOs are 0.84, 0.76, 0.66 and 0.60 respectively).
An important difference between TOs and other shapes relates to the particle orientations when the closest approach between two particles occurs (i.e.,
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Figure 5.8: Phase diagram of TO as a function of volume fraction Φ and confinement H*. Green shaded areas represent two-phase coexistence regions.
the shortest width of a particle). The shortest distance between any two parallel faces for cubes, TC4s and COs is the edge-length of the cube they were truncated from (and hence the distance between the square faces). However, for TOs, this minimal distance is that between opposite hexagonal faces. This means that for smallest values of H* studied (≈ 1.2), TOs tend to be aligned such that the hexagonal faces are parallel to the wall. We call this structure a HP phase, denoting a phase with hexagonal bond-order and with the hexagonal face parallel to the walls (see Figure 5.8). For H* = 1.4, however, we see the HT phase which also has hexagonal bond-order ( with high values of Ψ6), but the hexagonal face is tilted with respect to the wall. This is because, with the availability
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of more space in the Z-direction, TOs can increase their configurational entropy by exploring more orientations.
For 1.4 < H∗ ≤ 1.7, we see a buckled rhombic (BR) phase that is similar to those observed for the previous shapes. For (1.7 < H∗ < 2.0), we see a familiar two-layer square rotator (2S R) phase that eventually goes into a square phase (2S ). For (2.1 ≤ H∗ < 2.6), we see a single transition leading to a two-layer distorted rhombic (2RD) phase, that has high Ψ6 bond-orientational order and high global orientational order (see Figures B.6 and B.7 in Appendix B).
For 2.6 ≤ H∗ < 2.9, we see a novel phase previously not seen for other shapes. This phase has local coordination similar to the BCC structure of bulk TOs, however, it has the 111 plane of the putative BCC structure parallel to the wall. This creates hills and trenches in the structure since TOs now cannot pack space efficiently near the walls. Thus the particles separate into four different sets of heights along the Z axis), even though there is space for two full layers only. Such an arrangement is similar to the buckled phase seen for smaller H* values, except that it has now two full layers. Also, instead of the rhombic order previously detected in buckled phases, this phase has a square order (which mimics the local BCC bond coordination). We will refer to this phase as a two-layer buckled square phase (2BS ).
For H∗ ≥ 3.0, we first see the formation of a hexagonal rotator phase, 3HR (rotator phase with high Ψ6 order parameter but with little or no global orientational order), that upon further compression changes into a distorted rhombic phase (RD), with the number of layers increasing with H∗. Qualitatively, this reconciles well with the bulk-3D behavior for TOs. Although the maximum volume fractions are not close to 1 (as they would be for the space-filling phase
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Figure 5.9: Snapshots depicting the structure of TOs at representative confinement values: (a) Hexagonal parallel (HT ) phase at H*=1.2 and φ ≈ 0.53, (b) Buckled rhombic (BR) phase at H* = 1.6 and φ ≈ 0.56 , (c) Two-layer rotator square (2SR) phase at H* = 1.9 and φ ≈ 0.66, (d) two-layer buckled square (2BS ) phase at H*= 2.8 and φ ≈ 0.67. Particles are colored for ease of visualization and identification of layers only.
in the bulk), they steadily increase as the confinement is reduced.
Overall, the distinctive features of the phase behavior of TOs are likely due to their high truncation parameter (that makes them significantly different than cubes) while having low shape anisotropy (and low asphericity). The buckled square (2BS ) phase is unique to TOs and appears when the BCC motifs (that TOs would favor in the bulk) can fit inside the slit. Also noteworthy is that certain confinements cause TOs to form dense structure quite dissimilar from
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their preferable bulk BCC packing, such as a two-layer square rotator phase .
5.3.5 Remarks on global trends
Looking back at the details of the phase behavior of all the shapes described here, it is worthwhile to remark on some global trends we observe. These trends help us get a broad view of the interplay of spatial confinement and shape anisotropy. Further, connections to results already reported on similar systems help us understand our results in a wider context.
In particular, shapes studied here differ in asphericity (with cubes having the highest asphericity and TOs the lowest). In fact, the truncation process itself gives us a smooth way to go from a cube, to TC4, to CO to TO, to higher polyhedra and eventually to hard spheres (HS) and it is illuminating to understand their phase behavior in that progression. With decreasing asphericity (and hence increasing ‘roundedness’), we observe that the phase diagram gets progressively richer. This can be understood by the fact that with high asphericity, all except only a small set of configurations (for e.g. nearly parallel alignment to a wall) tend to have high entropic costs. With increasing roundness, higher and higher number of orientations are entropically favorable, which leads a richer variety.
Buckled phases are observed as we decrease the asphericity even a little bit (i.e. in TC4) and are seen in all shapes except cubes, and even in HSs[172]. From CO onwards, to a large extent, we see a progression of square-rhombic phases (2S → 2R → 3S → 3R.. etc.), similar to a 2 → 2 → 3 → 3 progression in HSs [172]). TC4s, on the other hand, still high in asphericity, only show a
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buckled and a square rotator phase, apart from the predictable distorted square (S D) phases.
Finally, we can also compare these results to previous calculations for polyhedra confined to a surface[94]. In the region 1 < H∗ < 1.8 or so, our system provides only enough room in the Z-direction for the particles in a monolayer to undergo rotations; such narrow-slit constraints can be conjectured to be similar to the flat-interface constraints for a monolayer system where the center of mass of the particles are confined to a stay on a flat surface. We indeed see a similar behavior for cubes in both scenarios. For TC4s on a flat interface, structures with both six-fold and four-fold orientational order were observed[94] as intermediates between the liquid and crystal phases. In our case, six-fold ordered structures are absent but we observe a buckled rhombic (BR) phase that has high Q6 order parameter at H* = 1.9. For COs, both narrow-slit and flatinterface constraints lead to the formation of a hexagonal rotator phase (HR) as well as a rhombic phase, which in our case is also buckled (due to the extra space along the Z axis). For TOs under flat-interface constraints, a hexagonal rotator (HR) phase is seen, which leads to rhombic phase upon further compression. In our case, we see a similar phase behavior with HP, HT and BR phases, all of which show 6-fold bonding symmetry.
5.4 Conclusions
We investigated the phase behavior of four representative polyhedral shapes from the truncated cube family, namely, cubes, TC4s, COs, and TOs, when confined inside a slit with hard walls separated a distance H*. The central aim
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was to explore the transitional phase behavior as the system goes from a quasi2D geometry (for strong confinement, small H*) to a quasi-3D bulk behavior (for weak confinement, large H*). The choice of shapes was motivated by the growing availability of experimental methods that readily allow synthesis of nanoparticles with these shapes and by the numerous theoretical and simulation studies that have predicted a rich bulk 3-D behavior for such systems.
While we found some common trends in all four shapes, there were many aspects of their phase behavior that were unique to each particular shape. The common trends help us understand the generic effects of geometric confinement. We observed a saw-tooth-like pattern in the ordered-state densest packing as a function of H* that is associated with how efficiently the particles use the available space while accommodating some integer number of layers. The phase behavior is particularly interesting as H∗ → 2, since the relatively large free space associated with the frustration of a a second layer allows the possibility of alternative structures that can maximize entropy. Except for cubes, we detect the formation of buckled phases for all the other shapes after H* goes over 1.5. For the H* values we investigated, however, the phase behavior does not fully converge to the bulk-3D behavior, although confinement effects weaken significantly beyond H∗ > 3. As the asphericity in the shapes is reduced, our results show features that approach those of the phase behavior observed for hard spheres (HSs) in parallel-plate confinement[172]. Our simulated phase behavior for strong confinement (so that only a monolayer is allowed) also show some similarities to that of freely rotating polyhedra whose centers of mass are confined to a flat surface[94].
The considerable effect of geometrical confinement on the phase behavior of
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truncated-cube type polyhedral nano- or micro-particles suggests that this provides an effective experimental route to engineer novel phases in these systems. Towards creating functional nanomaterials with tunable material properties, robust strategies such as shape anisotropy, shape bi-dispersity, enthalpic ‘patchy’ particles are often used. Our work adds to the body of work that shows that geometrical confinement can also effectively and controllably tune the particle structure and hence potentially other material properties associated with it.
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CHAPTER 6 CONCLUSIONS AND RELEVANCE TO EXPERIMENTS
Through the preceding chapters, we have attempted to describe the thermodynamic phase behavior of polyhedral nanoparticles as a result of bidispersity (binary mixtures) and parallel-plate confinement. Our primary objectives behind these studies were two-fold: (1) understanding and simplifying a vast array of factors that affect polyhedral self-assembly from a fundamental perspective, and (2) ascertaining guiding rules towards the design and synthesis of ordered superstructures from the point of view of experiments and technological applications. In this chapter, we give a concise summary of our findings and their relevance to experimental studies. We end with a discussion on currently prevalent ideas and open problems in the field.
6.1 Summary of key results
The main focus of this thesis is on the thermodynamic phase behavior of binary mixtures of polyhedral nanoparticles. We study these binary mixtures from a pure entropic perspective (treating them as hard particles) as justified by the details of many synthesis processes as well as from our motivation of isolating the physical effect of entropy alone. As explained in Section 1.5, this amounts to studying the interplay between packing and mixing entropy in binary mixtures of hard particles.
From the study of tessellated binary compounds, we find that the tessellation (space-filling) is a useful, but not so practical constraint in assembly of binary superlattices from polyhedra. Although the tessellated compound appears
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to be thermodynamically stable structure at high volume fractions, presence of kinetic traps is likely to prohibit the spontaneous assembly of binary mixtures on experimental timescales. Introducing small enthalpic interactions (that favor the tessellation) however, causes the binary mixture to spontaneously assemble into the compound phase. These favorable interactions could potentially be induced in experiments from DNA-grafting, electric charge or even from selective surface roughness. We discuss more about the kinetics of these assemblies in the next section.
From the study of generic, non-tessellating binary mixtures, we find that ∆ODP, the difference between the order-disorder transition pressures of the constituent shapes is a parameter that can predictively decide the mixture miscibility. Other simpler geometric parameters like ratio of volumes, circumradii or edge lengths do not fare as well as ∆ODP. We argue that the physical reason behind the importance of ∆ODP is the near-matching of the chemical potentials of the two shapes at the phase transition due to ODP-matching, as described in the Chapter 4.
From the intuition gained from the study of binary mixture miscibility, we propose a scheme for obtaining ordered binary superlattices from a purely entropic self-assembly of polyhedra. This scheme combines the use of polyhedral shapes exhibiting plastic crystalline mesophases with the ODP-matching (to maximize mixture miscibility through near-matching of chemical potential). Existence of mesophases likely reduces packing entropy thus allowing for ordered mixtures to be stable in spite of the entropic costs due to mixing. Our simulations using this strategy show robust ordered superstructures (MRMs) for a set of mixtures from the polyhedra of truncated cube family (see Chapter 4 for
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more details).
We also examine the phase behavior of monodisperse polyhedra under a parallel-plate confinement. The motivation behind this is to explore the effect of the spatial constraint on the polyhedral self-assembly (which for our pure entropic case, translates to entropy maximization under the confinement constraint). We observe that the confinement leads to a wide range of phases as a function of confinement width H*, with many phases not otherwise seen in bulk self-assembly. Particularly important is the region surrounding H*= 2, where the entropy maximization and incommensurability of bulk-lattice lead to a richer set of phases.
6.2 Relevance to experiments: study of kinetics
As stated earlier, a partial goal of this study is also to develop some guiding rules that help future experiments towards material design, particularly that of binary ordered superstructures. Without a set of guiding rules, a shot in the dark approach would be impractical considering the large number of control parameters and experimental conditions that can be explored. These guiding rules help us narrow down the region of likely candidates and control parameters for targeted design of colloidal superlattices.
The key results from our study (as stated in the preceding section) would of course be helpful in identifying likely candidates for superlattices. The shapes that are too dissimilar in size ( and more accurately, in ODP) are unlikely to be good candidates. The shapes involving no mesophases (plastic crystalline or liquid crystalline) are also likely to require more restrictive conditions to form
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ordered mixtures.
In case of tessellating compounds, we encounter a slightly more involved issue. Our simulations show that the self-assembly of compounds is hindered, at least in the case of hard particles, by a large free energy barrier. It is pertinent, however, to discuss the comparison between experimental and simulation timescales when we talk about kinetic barriers. It is the experimental timescale (rate) of a particular process (for e.g. a nucleation event), that determines the practical feasibility of a particular approach to self-assembly. If such ”high” barriers are surmountable within days rather than years, the approach might still seem viable.
To answer these questions about timescales, it is necessary to study the kinetics of such events. Most common set of methods employed towards that end include umbrella sampling (US)[116, 188], forward flux sampling (FFS) [120,189,190] and transition path sampling (TPS) [191,192] among others. These methods employ special techniques like non-Boltzmann sampling or sampling in the phase space of trajectories to investigate rare events more effectively. While US primarily gives the height of the free energy barrier, FFS allows us to calculate transition rates for the given process. For e.g. in FFS, a generalized reaction coordinate is used to divide the phase space by a sequence of interfaces ( λ0, λ1, ..., λN) that spans the nucleation process. The nucleation rate from the fluid phase A to the solid phase B is given by:

kAB = ΦAλ0 P(λN|λ0)
N−1
= ΦAλ0 P(λi+1|λi)
i=0
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(6.1) (6.2)

where ΦAλ0 is the steady-state flux of trajectories starting from the A state and crossing the interface λ0 in a given volume, and P(λi+1|λi) is the probability that a configuration starting at interface λi will reach interface λi+1 before it returns to fluid A.
One can then compare these rates to the experimental data in the following manner: The calculated rates can be represented in units of τ0, the time required for a colloidal particle to diffuse a distance equal to its diameter d (through regular diffusion process). τ0 can be connected to short time diffusion constant D0 (since τ0 = d2/D0) and long time self-diffusion coefficient DL (often calculated via light scattering experiments). For e.g. in case of hard spheres, following expression works quite well:

DL(φ) = D0

φ 1 − 0.58

δ

(6.3)

where φ is the volume fraction and the exponent δ takes values around 2.6

[193, 194]. The rates thus calculated from simulations, in these natural units,

convey how practical it would be to observe a process in experiments. For hard

spheres, a comparison between experimental and simulation data is available

in literature[142]. For the polyhedra studied here, these rates have already been

calculated for the isotropic-rotator transition via simulations[93]. We can take a

concrete example in that case. For a degree of supersaturation of 0.42 for TOs,

Thapar et al. [93] report a reduced rate of 5.8 × 10−7. The average expected time

for that process would be:

texp

=

tBr

×

V olume kr N

(6.4)

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where tBr, kr and N are the Brownian timescale, reduced rate and the number of particles, respectively. With a rough value of tBr of about 10−4 seconds for a particle around 50 nm in diameter, we can get texp to be of the order of about 1 second. Similar calculation for the case of tessellated compounds can yield rates that can give us the expected timescale for such a nucleation process, which would be of immense practical importance.
6.3 Currently prevalent ideas and open questions
Like most scientific studies, our attempt at understanding polyhedral selfassembly answers only a small subset of open questions currently plaguing researchers. In spite of the significant progress made in the field of polyhedral colloids during the last decade or so, there remain several areas where sufficient understanding is yet to be achieved. We discuss here some of the unsolved questions and currently prevalent approaches to address them (both from a fundamental and an application perspective), in the case of polyhedral colloids and more generally in the wider field of self-assembly:
1. Inverse methods for self-assembly: In colloidal self-assembly, there are typically two broad paradigms in terms of scientific inquiries. The first of them studies the forward problem in self-assembly, where starting from a present building block and a set of assembly conditions, it seeks to explore or predict the self-assembled structure. The second approach studies the inverse problem, where starting from a set of targeted structures and properties, it attempts to ascertain which building block(s) and assembly conditions would be required to assemble that structure.
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Such inverse methodologies hold great promise since they can potentially find interesting building blocks, interactions, and phase space regimes that allow for a given target structure, morphology or a property in a much faster way. Some of these methods called ‘reverse Monte Carlo’ methods[195–197] attempt to find a symmetric pair potential that reproduces the given structure factor or pair-correlation function data. However, others take a broader view and find interactions that can drive assembly of a variety of target morphologies in two or three dimensions[198–200] including Kagome, honeycomb, diamond and cubic lattices. Recent studies have also predicted structures with desired target properties using genetic algorithms for block-copolymers[201, 202] as well as artificial evolution based algorithms for granular materials [203].
2. Tailored interactions and multicomponent self-assembly: Rational material design approaches using some of the inverse methods mentioned above can prescribe the nature of interactions and assembly conditions required for a particular target structure. But this effort would be of little use unless such interactions could be realized in experimental systems. To that end, experimental studies have explored the effect of changing shape[37–39, 108, 204], surface coating[205–207], surface texture [115], particle-solvent interactions[208] and designing anisotropic interactions with DNA sticky-ends[209–211] and templating[59, 212].
As explained earlier in the thesis, more complex materials containing two or more components are often sought-after because of their wider range of applications compared to single species. A combination of the right stoichiometry, structural diversity and assembly conditions can potentially provide many possibilities of designing novel materials with targeted
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structural, electronic, catalytic or optical properties[100, 144, 213–216]. However, such assemblies tend to be challenging since they often need to have tailored interactions. Mixtures of many components also tend to get trapped kinetically (as our own studies[129] show) even more easily compared to single species because of the long timescales associated with swapping of two species once the solid has formed. This may seem to suggest that larger the number of building blocks, higher is the expected error rate (which can be quantified as a fraction of wrong motifs or the amount of structural defects) for the self-assembly.
However, recent results[217] demonstrated that as many as about 1000 different species of synthetic DNA strands, or so-called ‘DNA bricks’, can self-assemble into complicated, well-defined three-dimensional target structures. This result counters the presumed infeasibility of multicomponent self-assemblies. Further simulation studies by Reinhardt et al.[218] explain this apparent contradiction by the argument that the specificity of interactions (a given brick favoring a unique other brick) determines the neighbors and hence allows for error-free self-assembly. Given that Reinhardt et al. use no molecular detail of DNA strands, similar scheme can potentially be useful for colloidal super-atoms as well. Although these results look promising, it remains to be seen if a similar strategy would work for colloids as well.
3. Use of patchy particles: One of the principal approaches of generating diverse building blocks with colloids, is the use of ‘patches’ or directionspecific interactions. Inspired by directional interactions observed for model proteins that make viral capsids, patchy colloids have generated a lot of interest[219–221]. Further, several methods to synthesize patchy
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particles have now been established. Patchy colloids can be generated by selective deposition[222–224], colloidal lithography[225], microfluidic techniques[226–229], using colloidal clusters[211] among other methods. Number, size and position of the patches are among many control parameters that can decide the nature and course of the self-assembly process.
Among various types of patchy particles, Janus particles are one of the most commonly studied variety[230–232]. Due to their amphiphilic character, Janus particles can generally be easily anchored at the interface of oil and water or gas and liquid[233, 234]. Patchy particle assemblies have also shown formation of different structures like Kagome lattice[212] and have hinted towards formation of other open structures[235]. Experimental advances have also motivated many theoretical and numerical investigations[236–238]. Programmed assembly with DNA-functionalized patches[239] and magnetic patchy particles[240] open up even more possibilities for material design. Overall, patchy particles hold promise both as versatile building blocks, as well as model systems to study self-assembly in general.
4. Driven/directed self-assembly: The term directed self-assembly refers to the technique of controlling self-assembly processes via different experimental conditions, while still maintaining principal elements of spontaneous self-organization. Some of the methods used to gain such a control make use of electric or magnetic fields[241–244], curved interfaces[245] or flows or fields to direct the system to a desired phase. Directed self-assembly of block copolymer melts often makes use of electric fields[246, 247], magnetic fields[248, 249], templating or patterning[250, 251] or even shear[252–254].
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Use of flow fields like oscillatory shear is a powerful technique that can steer the self-assembly in different directions[255]. As we already discuss in this thesis, assemblies can often be hindered from getting to an equilibrium (and in many cases, desired) phase due to kinetic barriers. Flow fields often help in this scenario, since they can potentially drive systems to equilibrium. For e.g. colloidal glasses of spherical particles form closepacked crystals when subjected to oscillatory shear[256, 257]. In general, flow-fields can also cause structural rearrangements that can give rise to newer structures and a wider range of phases[167,258–261]. Together with external electric and magnetic fields, flow-fields have emerged as an important tool for directed self-assembly.
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APPENDIX A SUPPLEMENTARY INFORMATION ON MRM PHASE BEHAVIOR

A.1 Shape definitions

The shapes used for our main mixtures fall within the family of truncated cubes.

Truncated Octahedron (TO) is a platonic solid, while cuboctahedron (CO) is an

archimedean solid. The third shape, truncated cube, with truncation parameter

0.4 (TC4) is a cube with 80% of its corners cut-off. The detailed definition can

be found in the supplementary information of reference [92]. As described in

the main text, the relative ratios chosen were such that the ODP for each shape

was approximately the same (around 7.1). Such a ratio turns out to be the same,

if the TC4 and CO were to be carved out of the same cube, as is the case in

some experimental protocols[38, 39]. This is not true for the COTO and TC4TO

mixtures if the order-disorder pressures (ODPs) of the components are to be

matched.

Table A.1: Reference data for the shapes studied. ODP denotes the orderdisorder transition pressure which is the isotropic-rotator phase transition. MCP denotes the rotator-crystal transition pressure. For TC4 its two distinct edge-lengths are listed.

Shape Cuboctahedron Truncated Octahedron Truncated Cube (0.4)

Edge-length 0.707 0.4472 0.5657, 0.2

Volume 0.833 1.012 0.9147

ODP 7.1 7.1 7.1

MCP 14 14 14

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A.2 Isotropic branch of Equation of State (EoS) for different shapes

To explore some of the consequences of matching ODPs of the mixture components on the thermodynamics of the pure ordered phases at the ODP, Figure 1 shows the isotropic branch of the Equation of State (EoS) for each of the shapes discussed in the main text. The relative particle sizes of each shape correspond to ODP-matched conditions and, for convenience, the length scale for the reduced pressure P∗ is such that the ODP always corresponds to 7.1 (even for the spheres and cubes). Figure 1 plots the compressibility factor Z:

Z = βpV N

(A.1)

where β = 1/kBT . Note that for all cases Z →1 as P→0 (i.e., ideal gas behavior is approached). Z can be related to the residual chemical potential of the isotropic phase at the ODP via the thermodynamic integration:

µ∗ =

ODP (Z − 1) dP

0P

(A.2)

Since the isotropic phase coexists with the ordered phase at the ODP, it follows

that Eq. (2) also gives the chemical potential of the ordered phase. Further, since

for a hard-core system the residual entropy per particle is

S ∗/k = βPV − βµ∗

(A.3)

one can then readily estimate the S ∗ for the ordered phase at the ODP. By virtue of the significant similarity in the EoS isotropic-branches of the particles of interest (Fig. 1), the residual entropies of the pure-component ordered phases thus calculated have similar values for all cases.

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Figure A.1: Comparison of EoSs for different shapes at ODP-matched conditions for 0 < P∗ <ODP. C and S represent data for cubes and spheres respectively. Except for TC4, the EoS data for each of the shapes has been sourced from previously published results: TO[77], CO[87], cubes[88], Spheres[262].
A.3 Simulation procedure for binary mixtures with CO, TO
and TC4
Extensive expansion and compression Monte Carlo (MC) runs were carried out for the COTO, TC4TO, and TC4CO mixtures treating each component as a hard particle in the isothermal-isobaric (NPT) ensemble. A total of 2560 particles were used for each of the mixtures. A total of 3 × 106 MC runs per pressure step were used both for equilibration and production. Additionally, interfacial runs were used near the transition, to test the relative stability of phases. At pressures where the systems were ordered (as obtained in trial runs), the volume moves were allowed to be triclinic[77]. Swap moves between the two shapes were used to improve sampling[77]. Overlaps were checked using separating axis theorem[132]. A small hysteresis was observed in the transition pressures between compression and expansion runs; however, only results from the expansion runs are reported since they are typically closer to the equilibrium data
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Figure A.2: Equation of state for the equimolar COTO mixture
Figure A.3: Equation of state for the equimolar TC4TO mixture [124].
A.4 Equations of state for main binary mixtures studied
The equation of state (EOS) curves showing the variation of pressure as a function of volume fraction are presented for equimolar composition of COTO (Figure A.2), TC4TO (Figure A.3) and TC4CO (Figure A.4). In each of them, the MRM stability range is bound by vertical lines.
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Figure A.4: Equation of state for the equimolar TC4CO mixture
A.5 Ternary Mixture
As an additional test of the ODP-matching criterion to promote a mixed rotator mesophase (MRM), an equimolar ternary mixture of CO, TO and TC4 was also simulated with a total of 3000 particles. An approximate equation of state (EOS) for this mixture and a representative snapshot of the ternary MRM phase are shown in Figures A.5 and A.6 respectively. The EOS shows that as in the case of the binary mixtures, the MRM phase is stable for a considerable range of volume fractions (≈ 0.51-0.68). While the MRM phase separates into two solid phases above P* ≈ 18, the CO-TC4-rich solid phase further separates into two solid phases (each rich in the individual components) above P* ≈ 30. However, that region is not shown in Fig. A.5 nor an attempt was made to outline the full ternary phase diagram since the whole region above the MRM stability is only of marginal interest and has only been tentatively characterized here. Detailed analysis of the phase behavior and the structure of the ternary MRM phase is also beyond the scope and focus of this work.
123

Figure A.5: Equation of state for the ternary mixture. Above P* ≈ 18, the mixture phase-separates into TO-rich and CO-TC4 rich solid phases.
Figure A.6: Snapshot of MRM phase in the equimolar ternary mixture of CO, TO and TC4 , with a total of 3000 particles at P*≈ 11.2
A.6 Simulation details for mixtures of spheres and cubes
These mixtures were simulated in a semigrand isobaric-isothermal ensemble, where the total number of particles was fixed at N=1000 for cube-rich phases and N=864 for sphere-rich phases. The composition of the system is not fixed but allowed to fluctuate in response to a specified value of the chemical potential difference ξ = β(µ2 − µ1), letting species 1 and 2 correspond to cubes and spheres, respectively. The dimensionless osmotic pressure is given in reduced units as P∗ = βPσ3, where σ is the cube edge length. Runs for any given system and state involved 3 × 106 MC cycles for equilibration and 3 × 106 MC cycles for production. Each MC cycle consisted on average of N translational, N rotational (applied to cubes only), N/20 swap, N/2 mutation, and 4 volume move
124

attempts. Swap moves are similar to those used for the main mixtures. Mutation moves allow composition equilibration and involved picking a particle at random and attempting to mutate its original identity into another species type, keeping its old position (and for a sphere → cube mutation, choosing a random orientation for the cube). All trial moves are accepted according to the Metropolis criterion which rules out overlaps as described in [148]. Acceptance rates for mutation moves varied significantly with system pressure and the size ratio between components; these become particularly small [O(10−6] for high pressures of the most asymmetric mixture (for cubes and spheres with equal circumradius) which sometimes necessitated the repetition of runs to try to attain converged compositions.
Pressure-composition coexistence lines were mapped stepwise by a recent variant of the Gibbs-Duhem integration method [148]. Two sets of simulations were performed, one started at the known pure-sphere solid-liquid coexistence conditions and the other at the known pure-cube solid-liquid coexistence conditions. Therefrom (and for each such set), the pressure was changed incrementally, the corresponding coexistence ξ value calculated and new near-coexistence simulations conducted to allow the stepwise process to continue. Once the eutectic pressure is reached, the two sets of simulations then become only one (as the liquid phase disappears); above this pressure, only one set of simulations is needed for mapping out the phase coexistence between the sphere-rich solid and the cube-rich solid.
125

A.7 Metrics for the analysis of configurations

A.7.1 P4, Q4, Q6 order parameters

The positional and orientational order in the simulated configurations was char-

acterized via the calculation of the P4, Q4, Q6 order parameters [86, 113]. The Q4

and Q6 bond-order orientational parameters are defined as:

1

Ql =

4π 2l +

1

 

+l

2 |Q¯ lm(r)|2

−l

(A.4)

where Q¯lm(r) is given by

Q¯ lm(r)

=

1 Nb

Ylm(r)
bond s

where Ylm(r) are spherical harmonics for the position vector r.

(A.5)

Figures A.7, A.8, and A.9 show the plots of P4 and Q6 order parameters for the two components of each of the three mixtures COTO, TC4TO and TC4CO respectively. These show that in the MRM positional order (Q6) is always high while the orientational order (P4) is low or intermediate, but in both cases order tends to increase with pressure.

A.7.2 MRM structure determination

The MRM structure was characterized by analyzing the distributions of two

local bond-order orientational parameters q¯4 and q¯6 as defined by Lechner et al.

[263]

1

q¯l(i) =

4π 2l +

1

 

+l

2 |q¯lm(i)|2

−l

(A.6)

126

Figure A.7: Variation of P4 and Q6 order parameters for COs and TOs in an equimolar COTO mixture. The MRM phase boundaries are marked by the dashed lines. Order parameters are for specific species before phase separation and for the two phases thereafter.
Figure A.8: Variation of P4 and Q6 order parameters for TC4 and TO shapes in an equimolar TC4TO mixture. The MRM phase boundaries are marked by the dashed lines. Order parameters are for specific species before phase separation and for the two phases thereafter.
127

Figure A.9: Variation of P4 and Q6 order parameters for TC4 and CO shapes in an equimolar TC4CO mixture. The MRM phase boundaries are marked by the dashed lines. Order parameters are for specific species before phase separation and for the two phases thereafter.

where Q¯lm(r) is given by

q¯lm(i) =

1

Nb(i)
qlm(k)

Nb(i) k=0

(A.7)

where the complex vectors qlm(i) for each particle and all its Nb(i) neighbors are

averaged to get q¯lm(i) which makes the distributions more disjoint for the vari-

ous possible crystalline structures. These local order parameters form the dis-

tribution vector νˆ, obtained by stacking together local q¯4, q¯6 distributions. This

distribution vector is then used to estimate fractions of various possible crys-

talline orders in the MRM by comparison to distribution vectors for reference

BCC, FCC, HCP and liquid phases. The distributions for the reference phases

are plotted in the Figure A.10.

The fractions of different structures are calculated by minimizing ∆2 in the following equation:

∆2 =

2
νˆmeso − ( fbccνˆbcc + f f ccνˆ f cc + fhcpνˆhcp + fliqνˆliq)

(A.8)

128

Figure A.10: Plot showing distributions of different order parameters for reference phases

subject to constraint:

fbcc + f f cc + fhcp + fliq = 1

(A.9)

The variation of these fractions in the MRM for COTO, TC4TO and TC4CO mixtures is shown in Figures A.11, A.12, and A.13. Note an apparent anomaly at the lower end of the MRM phase for COTO and TC4CO, where a high fraction of BCC occurs. These structure fractions are not expected to be discriminatory in the low-pressure region since in the early stages of MRM formation, the overall order itself is quite low.

129

Figure A.11: Variation of the fraction of different crystal structures in the equimolar COTO MRM. The bounds of the MRM are shown by dashed vertical lines.
Figure A.12: Variation of the fraction of different crystal structures in the equimolar TC4TO MRM. The bounds of the MRM are shown by dashed vertical lines.
A.7.3 Orientational scatterplots
The extent of orientational order in the mixture configurations can be visualized via orientational scatterplots. They are generated by plotting the three orientation axes of each particle on the surface of a unit sphere. A uniform pattern is indicative of orientational disorder, while a pattern containing (sharp or diffuse) clusters is indicative of orientational correlations.
130

Figure A.13: Variation of the fraction of different crystal structures in the equimolar TC4CO MRM. The bounds of the MRM are shown by dashed vertical lines.
Representative scatterplots for each shape in the COTO, TC4CO and TC4TO mixtures are shown in Figure A.14. One can see how the orientational disorder at low pressures in the liquid phase (leftmost panels) slowly gives way to an intermediate-degree of clustering in the rotator phase (middle panels), eventually leading to highly patchy scatterplots in the solid phase (rightmost panels). Also note that in the COTO mixture the TOs are considerably more ordered than COs at a given pressures. Such a disparity between components is not observed in the other two mixtures, likely due to the greater shape similarity between the components.
131

Figure A.14: Orientational scatterplots for individual shapes in the main three mixtures at equimolar composition at representative pressure regimes. Starting from left, the first column shows the isotropic regime; the second and third columns show the low- and high-pressure ends of the MRM regime, respectively, while fourth column shows the solid phase (which is rich in the indicated component) in the phase-separated regime.
132

APPENDIX B SUPPLEMENTARY INFORMATION ON POLYHEDRA IN PARALLEL
PLATE CONFINEMENT
We use this section to provide additional details and plots regarding some of the phases described in the main text. In the plots to follow, we use the symbols P41 and P42 to denote the P4 values associated with the two directors.
B.1 Cubes
Figure B.1: A Plot showing the equation of state as well as various order parameters including P41, P42, Ψ4 and Ψ6 for cubes at H* = 1.1. All order parameters are scaled so that they have a maximum value of 1.0. The region between φ = 0.58 to 0.64 has significant particle orientational order and intermediate four-fold bond-orientational order.
As discussed in the main text, we did not observe cubatic mesophase as seen in the 3D case[77, 86]. However, for certain H* values (particularly, those close to H*=1), we see small regions of tetratic order. These regions are characterized by significant particle orientational order and intermediate four-fold
133

bond-orientational order. As an example, we show plots for H*=1.1 and H*=1.3 (see Figures B.1 and B.2)
Figure B.2: A Plot showing the equation of state as well as various order parameters for cubes at H* = 1.3. All order parameters are scaled so that they have a maximum value of 1.0. The region between φ = 0.45 to 0.52 has significant particle orientational order and intermediate four-fold bond-orientational order.
B.2 TC4s
For TC4s at H* = 2.0, we see a square rotator (2SR) phase at intermediate volume fractions. We can see the evidence for it in the order parameter plots (see Figure B.3), as the system gains significant four-fold bond-orientational order at φ = 0.39 while being orientationally disordered. At φ ≈ 0.47, it attains moderate orientational order thus leading to a 2S square phase.
134

Figure B.3: A Plot showing the equation of state as well as various order parameters for TC4s at H* = 2.0. All order parameters are scaled so that they have a maximum value of 1.0. The region 0.39 < φ < 0.47 is the intermediate 2SR rotator phase.
B.3 COs
Within the confinement range 1.5 < H∗ < 1.7, we observe hexagonal rotator (1HR) phase for COs. This 1HR phase is characterized by high values of Ψ6 signifying six-fold bond-orientational order but low orientational order (see Figure B.4). 1HR phase eventually gives way to buckled rhombic BR phase.
Such a two-step phase transition is also present for larger values of H*. Particularly, in the regime 2.6 < H∗ < 2.8, first transition from isotropic to 2HR, characterized by sharp increase in Ψ6 values appears to be of the first order while the second transition which marks a steady increase in the orientational order is continuous (see Figure B.5).
135

Figure B.4: A Plot showing the equation of state and various order parameters for COs at H* = 2.0. All order parameters are scaled so that they have a maximum value of 1.0.
Figure B.5: A Plot showing the equation of state as well as various order parameters for COs at H* = 2.8. All order parameters are scaled so that they have a maximum value of 1.0. This plot shows two-step transition for COs, first from isotropic to 2HR and then from 2HR continuously to 2RD.
136

B.4 TOs
Within the confinement range 1.7 < H∗ < 2.0, we see familiar square-rotator phase that eventually transitions into a square phase. This two-step transition can be seen in Figure B.6. The system first gains four-fold bond-orientational order at φ ≈ 0.4 (as shown by Ψ4 values) and at φ ≈ 0.56 gains the orientational order as well (as evidenced by P4 values).
Figure B.6: A Plot showing the equation of state as well as various order parameters for TOs at H* = 1.9. All order parameters are scaled so that they have a maximum value of 1.0. This plot shows two-step transition for TOs, first from isotropic to a square rotator 2S R and then from 2S R to 2S square phase.
On the other hand, in the region 2.1 < H∗ < 2.6, TOs show only a single-step transition from isotropic to two-layer distorted rhombic (2RD) phase. As can be seen in Figure B.7, TOs gain both particle orientational order and four-fold bond-orientational order to transition into 2RD phase.
137

Figure B.7: A Plot showing the equation of state as well as various order parameters for TOs at H* = 2.2. All order parameters are scaled so that they have a maximum value of 1.0. This plot shows single-step transition for TOs from isotropic to a distorted rhombic 2RD phase.
138

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