Chapter 1
Theoretical Studies and Tailored Computer Simulations in Self-Assembling Systems: A General Aspect

Zihan Huang and Li-Tang Yan

Key Laboratory of Advanced Materials (MOE), Department of Chemical Engineering, Tsinghua University, Beijing, China

1.1 Introduction

Self-assembly-a governing principle by which materials form-is the autonomous organization of matter into ordered arrangements [1, 2]. It is typically associated with thermodynamic equilibrium, the organized structures being characterized by a minimum in the system's free energy, although this definition is too broad. Self-assembling processes are ubiquitous in nature, ranging, for example, from the opalescent inner surface of the abalone shell to the internal compartments of a living cell [3]. By these processes, nanoparticles or other discrete components spontaneously organize due to direct specific interactions and/or indirectly, through their environment. Self-assembly is one of the few practical strategies for making ensembles of nanostructures. It will therefore be an essential part of nanotechnology. Self-assembly is also common to many dynamic, multicomponent systems, from smart materials and self-healing structures to netted sensors and computer networks. In the world of biology, living cells self-assemble, and understanding life will therefore require understanding self-assembly. The cell also offers countless examples of functional self-assembly that stimulate the design of non-living systems [4, 5].

Self-assembly reflects information coded (as shape, surface properties, charge, polarizability, magnetic dipole, mass, etc.) in individual components; these characters determine the interaction among them. The design of building blocks that organize themselves into desired structure and functions is the key to applications of self-assembly [2]. Much of materials science and soft condensed-matter physics in the past century involved the study of self-assembly of fundamental building blocks (typically atoms, molecules, macromolecules, and colloidal particles) into bulk thermodynamic phases [6]. Today, the extent to which these building blocks can be engineered has undergone a quantum leap. Tailor-made, submicrometer particles will be the building blocks of a new generation of nanostructured materials with unique physical properties [7-11]. These new building blocks will be the "atoms" and "molecules" of tomorrow's materials, self-assembling into novel structures made possible solely by their unique design [2]. For example, patchy particles consisting of various compartments of different chemistry or polarity are ideal building blocks of potentially complex shapes with competing interactions that expand the range of self-assembled structures beyond those exhibited by traditional amphiphiles such as surfactants and block copolymers [9]. By controlling the placement of "sticky" patches on the particles, assemblies can be made that mimic atomic bonding in molecules [8]. This greatly expands the range of structures that can be assembled from small components. Extension of the principles to particles of alternative compositions (such as those made from noble metals, semiconductors or oxides) will allow optical, electronic and catalytic materials to be coupled in previously impossible architectures that have potentially new emergent properties. Playing tricks with designer "atoms" also includes the shape of the building blocks [12]. For instance, the local curvature of dumbbell-shaped nanoparticles can be harnessed to control the ionization state of a molecular layer adsorbed on their surfaces and the self-assembly patterns of the particles [13].

Understanding the relation between building blocks and their assemblies is essential for materials design because physical properties depend intimately on structure, which however poses many challenges if considering complex thermodynamic and kinetic behaviors involved in the assembling processes. Indeed, a priori prediction of hierarchically assembled structures from a desired building block requires an in-depth understanding of the delicate balance between entropic and enthalpic interactions [14-17]. Central to this issue is exploring entropy-driven structural organization, because entropy keeps springing non-intuitive findings in the manipulation of the self-assembly of nanoparticles and the structural formation of soft matter systems [14]. On the other hand, directed self-assembly using a template or an external field may also give rise to novel ordered non-equilibrium structures, free from the constraints of entropy maximization, and hence these systems can "reside" in a state of local equilibrium within the global free energy with low entropy states often characterized by complex spatial or coherent spatiotemporal organization [18]. In this case, identifying the possible structures at metastable states is important for controlling the formation of structures or patterns. Not surprisingly, many advances have been made in theoretical models and simulation approaches to predict and analyze structures, dynamics and properties of self-assembling systems; computer simulations offer a unique approach to identify and separate individual contributions to the phenomenon or process of interest [19, 20].

However, the theoretical and computational research of self-assembling systems is far from trivial. These many-body systems cover variations in relevant time and length scales over many orders of magnitude. The assembled structures and macroscopic properties of materials are ultimately to be deduced from the dynamics of the microscopic, molecular level, implicating a lot of demand for new simulation techniques and theoretical approaches. From the computational point of view, the key is to develop methods capable of reaching time and length scales much larger than those accessible by brute force computer simulations on the atomic level [21]. The feat is not a simple one, since it requires a major effort over a wide range of activities, including the development of coarse graining techniques, novel simulation methods and ways to link the different regimes to each other. Even with these challenges, theory and simulations have proven invaluable and indispensable in studies of self-assembling systems, including applications in numerous directions such as development and examination of new principles, predictive science and computer design of complex building blocks, suggesting guidelines of programmable assembly, and exploring entropy interaction in various assembling systems, etc. The purpose of this chapter is therefore to introduce the general aspects of the development and applications of theoretical approaches and computational modeling in self assembling systems, focusing on basic and emerging principles.

1.2 Emerging Self-Assembling Principles

1.2.1 Predictive Science and Rational Design of Complex Building Blocks

Predicting structure from the attributes of a material's building blocks remains a challenge and the central goal for materials science. Here we introduce the rational design and predictive science of two emerging and important building blocks for superstructure construction through self-assembly, that is, polyhedral particles and particles that can self-assemble into helical structures with chirality.

At present, a major focus in material science is to engineer particles with anisotropic shapes and interaction fields that can be self-assembled into complex target structures [1, 2]. Assemblies of anisotropic particles undergo order-disorder transitions involving changes in both translational and rotational degrees of freedom and can lead to phases with partial structural order or "mesophases" [22, 23] such as crystals, plastic crystals and liquid crystals. These ordered assemblies have distinctive electronic, optical and dynamical properties and are highly desirable for fabrication of advanced electronic, photonic and rheological devices [24]. Although numerous theoretical [25, 26] and experimental [27, 28] studies on mesophase behavior of particles with anisotropic shapes have been reported, a roadmap marking out the most probable mesophases that could be formed by constituent particles with particular geometrical features remains incomplete. Exploring such relations will translate into a deeper understanding of the phase behavior of colloidal systems with different particle shapes. The simulation prediction of a dodecagonal quasicrystal with tetrahedra demonstrated the unexpected complexity that could be achieved for particles solely with hard interactions [29]. Escobedo and Agarwal [30] carried out detailed Monte Carlo simulations of six convex space-filling polyhedrons to demonstrate that translational and orientational excluded-volume fields encoded in particles with anisotropic shapes can lead to purely entropy-driven assembly of morphologies with specific order and symmetry. Their simulations reveal the formation of various new liquid-crystalline and plastic-crystalline phases at intermediate volume fractions. They further propose simple guidelines for predicting phase behavior of polyhedral particles: high rotational symmetry is in general conducive to mesophase formation, with low anisotropy favoring plastic-solid behavior and intermediate anisotropy (or high uniaxial anisotropy) favoring liquid-crystalline behavior.

Recently, a more refined structure prediction of polyhedral particles has been attained by Glotzer et al. [31] through investigating 145 convex polyhedra whose assembly arises solely from their anisotropic shape. Their simulations...