Engineering Matter Across Scales
Written by Arthur L. Robinson
The ability to design and make the perfect material with just the right properties to do what we want, how we want, and when we want is the holy grail of materials research says Sharon Glotzer of the University of Michigan. Such “materials on demand” require control over thermodynamics, kinetics, nonequilibrium behavior and structure across many length and timescales. In her Fred Kavli Distinguished Lecture in Materials Science during Monday evening’s Plenary Session, Glotzer took it upon herself to demonstrate how atomic and molecular crystal structures—made possible by chemical bonds at Angstrom scales—can be realized with noninteracting nanoparticles and colloids via entropic bonds at nanometer to micron scales. Beyond their importance in understanding and engineering the self-assembly of colloidal crystals and nanoparticle superlattices, Glotzer says, the fact that some of the most complex structures in metallurgy and in molecular crystals can be realized without explicit attraction of any kind, reveals fundamental insights into what is needed to engineer matter across scales.
Physical matter is held together by chemical bonds (ionic, covalent, hydrogen, metallic, and so on). It is a combination of quantum theory, which describes interatomic interactions, and statistical thermodynamics, which governs free-energy minimization, that determine all possible crystal structures in nature. Stable crystals can be predicted if we know all the interatomic forces and can minimize the free energy subject to thermodynamic constraints. In principle, the chemical bonding structure of any set of N atoms can be computed, with varying degrees of accuracy depending on the approximations used.
Shifting to soft matter, Glotzer explains that the entities playing the role of “atoms,” being big and complex, are different. Examples are dendrimers molecular, DNA, proteins, micelles, nanoparticles, including those functionalized with ligands, and viruses. Scores of three-dimensional crystal structures can self-assemble in solution from soft-matter building blocks. Most are isostructural to atomic crystals, though with larger lattice spacings. In every case, interparticle interactions combined with thermodynamics dictate crystal morphologies.
Particle shape plays a big role in dictating colloidal crystal structure because the anisotropy can create an effective “valence” that dictates the number and bond orientation of neighboring particles. Clathrates, structures consisting of polyhedral cages with large pores that can be used for host-guest chemistry, represent a challenging target for colloidal assembly. Here, particles of sizes ranging from a few nanometers to a couple of microns self-assemble in solution to form crystals where the “atoms” are replaced by particles made of atoms. Typically these particles are metals like Au or Ag, semiconductors like CdTe or CdS, or polymer like PS or PMMA, and functionalized with molecular organic ligands or DNA. The particles can be charged or neutral, be magnetic or not, spherical or rod-like or polyhedral. Particles can interact via electrostatic interactions mediated by the solvent, van der Waals interactions, magnetic dipole interactions, h-bonds between ligands, and excluded-volume interactions. Regardless of what interactions are present, they conspire to produce net interparticle repulsion and attraction that—combined with thermodynamics—dictate the preferred arrangement of particle positions and orientations. Today these colloidal crystals can be predicted, designed, and synthesized.
Very counterintuitive, says Glotzer, is that even in the absence of any explicit interparticle interactions, colloidal particle can form crystals due solely to particle shape and excluded-volume interactions. In these cases, free energy minimization is the same thing as entropy maximization. Entropy alone can drive self-assembly of an incredible diversity of colloidal crystal structures and with extraordinary complexity, both with and without atomic or molecular analogues. Upon crowding, hard particles organize to maximize the system entropy by maximizing the number of allowed microstates. Lots of questions remain: What else is possible with entropy alone? What crystals structures are not possible with entropy alone; if not, why not? What about multi-atomic systems of shapes that are not necessarily hard? To what extent is entropy helping to order nanoparticles into colloidal crystals? Can we engineer entropy to engineer target crystals?
Finishing up, Glotzer described entropic bonding as a process that selects for the set of interparticle orientations and positions that maximizes system entropy in analogy with chemical bonding as a process that selects for the set of interatomic orientations and positions that minimizes the total energy among atoms. In this analogy, entropy creates an emergent “valence.” With an entropic analog to the Schrödinger equation based on “shape orbitals” that can be placed on a lattice and optimized for maximum overlap to find the bonding orbital and free energy of a unit cell, Glotzer and colleague Thi Vo developed a predictive microscopic theory of entropic bonding based on a pseudopotential between a quasiparticle and a hard shape. They replaced the electron probability density (wave function) in the Schrödinger equation with the quasiparticle probability density, where quasiparticles serve as a proxy for an effective entropic attraction between shapes in a Schrödinger-like equation. Applying their theory to a hard-shape system of two hard cubes in a box at a density of 0.55, they found their model predicted a simple cubic “lattice” that would have lower energy than fcc or bcc, whereas for a system of truncated tetrahedra, the preferred lattice would progress from that of a quasicrystal to diamond to -Sn to bcc as the degree of truncation increases. These predictions match those from molecular-assembly simulations made in 2012.
For takeaways, Glotzer noted that entropic forces can be directional, effect valence, and act as bonds; there is now a predictive microscopic theory of entropic bonds; and this allows the use of approaches used for atomic crystals to be used for colloidal crystals.
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